Whew, where do I start?
OK. 422707....opposed twin cylinder. Your engine with a type number of 1510, is the Briggs Twin II.
This engine has a single ignition magneto (coil) that fires both cylinders simultaneously, so if you used any of the Vanguard wiring in the magneto kill circuit, that could be why the engines runs on. The Vanguard V-twin uses a "diode pair" wiring pigtail that electrically separates the "2 magneto" ignition system coils in order to allowing killing both mags using a single kill wire for stopping the engine.......without the 2 magnetos being able to "influence" or interfere with each other (regarding sparking).
The Opposed Twin II needs just a plain piece of wire connecting the magneto kill terminal to the kill circuits of the mower.
And rest assured, the magneto DOES NOT use any 12V power from the mower to make spark, it is self powered by the flywheel magnets. Running 12V power to a magneto is a sure way to blow the magneto's brains out.....permanently.
You are probably going to need to pull off the flywheel cover and identify that the Red and Black wires go to the stator, and determine if there is a wire connected to the kill wire spade terminal on the magneto.
If the stator wires on your OPP Twin II are Red & Black, then the engine is configured with the "Dual circuit, unregulated alternator" This rendition of alternator only has DC output of 2 to 4 amps for charging battery, and there is a separate AC only circuit which can be used to power lights (but it will not charge battery). The max AC output is 5 amp. If the Snapper has a manual PTO, this alternator will serve OK, but if there is an electric PTO clutch in the picture, you are woefully underwhelmed in the electrical department.
The DC circuit does not utilize a regulator (did we say that already?) but there is a diode to RECTIFY the current and that diode must be present, and oriented correctly in order to charge battery.
If you don't have a service manual for the Opp Twin II, e-mail me and I'll send you one (PDF document).

Yes it has an electric PTO.

I see in the parts manuals for the engines. That they potentially share the same alternator. From what I can tell from the parts pictures for the vanguard it looks like I had a 10 amp alternator. In the parts pdf for what we will now call the twin II (pars list model series 422700)I see the same 10 amp "696458 Alternator (10 to 16 amp regulated)" found in the vanguard parts list. Now if I follow you correctly the alternator in the twin II is just not gonna be able to power the battery and the electric PTO. So this makes me wonder if I could take the alternator out of the vanguard and install it in the twin II.

And one last wonder at this time... if the engine doesn't need battery power to run. Why is it that when the mower died my battery was totally dead. I charged it up and it now starts and runs fine again. Could I be looking at another problem?

The stator will interchange, but you also need the flywheel for the higher capacity charging system to go with it. It has "more" magnets.

I don't know if the flywheels will interchange "physically" or not, but you can look, since you have them both.

Unless I have erred, the image below avers that stator part # 696458 is used on both the Vanguard (series 350000) and the Twin II (series 420000). It also states that when this stator is used with a flywheel having "small magnets" the max output would be 10 amp.
I believe the OEM flywheel on that Twin II has "small magnets" since it came with the 2-4 amp alternator OEM.
Looking at the IPLs for both engines I find that voltage regulator part # 691185 is used on both engines when configured with stator # 696458.

What I'm not able to get from anywhere (so far) is how the heck would you go to larger magnets (if desired) because I find no distinctions being made in any IPLs regarding "different flywheels with different magnets??
And if one goes by part # alone.......a Vanguard flywheel WILL NOT interchange with a Twin II (or vice versa).

{{gwi:327134}}

Fly wheels will not swap. However I don't see any different flywheel options in the parts list. So that leads me to believe that I can get away with a stator/alternator swap. Or do they put less magnets on at factory but when they sell you parts have a one standard that will work?

Found what I believe to be the kill wire on the magneto. Just to be clear I DO NOT hook this up to 12V+ but if I hook it to ground it will kill the motor. Yes?

Lastly, would it be feasible to make a rectifier to put on the ac out and run that to the battery too? That would be a grand total of 7 to 9 amps by my calculations (err addition). Enough to power the electric PTO? I guess this would be single phase so I would need what two diodes one to ground and the other to positive?

Hmm two possible solutions to my problem both solutions a bit dubious.
Thanks for all the help.
I feel I'm slowly but surely getting closer to a working mower.

***"Found what I believe to be the kill wire on the magneto. Just to be clear I DO NOT hook this up to 12V+ but if I hook it to ground it will kill the motor. Yes?"***
Correct on both presumptions.
And, if my presumptions are right (in my other post above)
The stator from the Vanguard installed on the Twin II, using the Twin II flywheel will give you a max output of around 10 amp. (using the regulator from the Vanguard too)
I supposed you "could" rectify the AC output and use it for DC to add to the total, but diode orientation would be crucial or you would at the very least fry the diode (a diode installed backward in a circuit that connects a stator to a battery IS A DEAD SHORT TO GROUND, mucho smoko)

Well not that I don't think I could make the rectifier circuit right. Been playing with three phase rectifiers most of my working life. However it now seems to me that the safest bet is to go for the stator swap. Wish me luck pulling the flywheels. Proper pullin' tool comes in tomorrow. Flywheel wrench on the other hand...*points over at other thread*

Here is a link that might be useful: other thread

That's the way I would go.
Most likely, you'll have enough charging to take care of the battery.

I don't know if that mower has headlights and/or electric PTO.
Possibly, you may not generate enough to operate both at the same time, but let us know how it works out.
I'm rather curious how much "charge" you can get.

Headlights and electric PTO. Ill try and post back when I get it all together the stumbling block right now is the flywheel nut on the twin II. From what I'm seeing in the parts manual I see no reason it shouldn't do the 10A that the Vanguard did. Even the guys at the parts shop said it should work if there were a magnet difference it should show in the parts manual as a different flywheel option. On the parts manual for the Twin II there is only one flywheel to choose from.
I have to second mownie on the 10 to 16 amp question. There is absolutely no distinction parts wise.

If headlights turn out to be a necessary utility, you can just use the AC output and separate the headlights circuitry from the DC electrical system. Incandescent lamps DO NOT gag when you feed them AC. Incandescent lights used in a house work on AC and many Lawn & Garden Tractors come OEM with AC driven headlamps. The ONLY drawback with AC headlights is the brightness varies with engine RPM. Low RPM = dim lights, high RPM = nice and bright lights.

All updates concerning this engine swap will end up here ... not in the big ole nut thread.
It looks like Bill K was right. Just talked to a B&S tech. The 10 amp had bigger mags on the flywheel despite the one part number. So I'm not quite pushing enough power even with the stator swap. It does however push enough to work the PTO clutch maybe with a little battery help I really don't know yet for sure but it has done a fine job mowing half the acre. Only reason I know for sure its not pushing enough is when I engage the PTO the battery charge light comes on too. According to the B&S tech it should work fine for me if I put a charger on it after using it. Which IMHO is a small price to pay for not having to go out and buy a new throwaway mower.

As for my measurements on the electrics they came up wonky as all get out ... I'm beginning to wonder about the accuracy of my multimeter. 2amps at .5 volts? Ill give it another test later and hopefully get a better reading. Gonna get some alligator clips for the multimeter so I know that I am making contact well. The little pokie probes are unreliable in my book unless I can actually stick them in something. For now I must go and get another bolt to hold the steering linkage that I apparently didn't bolt down well enough after dropping the engine in.