As a side note, the Toro website lists this engine as a 12H802-1776-B1. The B&S website lists no replacement engine for the 12H802 type, and doesn't even list the 1776 model (not too surprising, since it was custom for Toro.)

I don't care if it is identical to the original, as long as it will mount my blade and run it at the correct speed with enough power to mulch my Fescue.

I assume that you've covered your bases on maintenance. SAE 30 oil, checked the air filter for oil contamination, not running rich etc. How much oil is it using? If you ran it a tad under full, does it still smoke?
The link below has a similar engine to which you could see if it will fit your needs. The IPL will indicate the differences if any. The Toro has the larger blade bolt 7/16" instead of 3/8" or Replacement Shortblock 792742 & Crankshaft Number 696009

Here is a link that might be useful: BS 4 Toro

Okay, another update (after some more Googling)... found another B&S Service Engine guide that wasn't offered to me by B&S's website.

It states that it can be replaced by a 12J802-2369. That seems to go online for $230-ish plus shipping. The guide also mentions I can use short block 696276, which I found at one store for a puny $60 + $25 shipping.

If we make the bold assumption that that site is legit, how much work is it to swap out a short block for a B&S Quantum? And are the parts not included likely to not long outlast the original short block? What is included in the short block anyway?

At that price it seems almost prudent to keep one on hand! The rest of this mower just doesn't have much on it that can break other than cables and belts. The transmission is just a gearbox without even an engagement mechanism to break. (I love the simplicity of the Personal Pace setup...) The deck is cast aluminum, so it's not going anywhere any time soon.

Here is a link that might be useful: 12H800 replacement engine guide

Along with tomplum I think there are many procedures/possible fixes to go through before pronouncing it an oil burner. Please go through his list of possibles and make sure all of those suggestions are examined. Sometimes the piston rings get a carboned-up from running rich and 'freeze', not being able to expand as they should. Some more basic questions; are you seeing blue/white smoke or black/blue? Any black in the exhaust indicates too much fuel. This 'washes' the oil off the cylinder and burns it. Bad fuel can also cause smoke. You mentioned 'getting things unstuck' for the first start of the season - that is not a normal condition, particularly for an aluminum cylinder without an iron sleeve. If the engine was not properly prepared for winter storage and was stored in a humid location, the rings could develop rust and still stick to the cylinder, causing some pitting which might lead to oil consumption. After determining there is nothing wrong with the fuel and you're using SAE30HD oil (not overfilled) along with all the other suggestions, I'd be inclined to run an internal engine cleaner through it to see if the rings will respond. Seafoam is a gentle cleaner that can work over time, and you have nothing to lose by trying it. You would use it in the oil and fuel.
A 'short block' is the basic power head; complete reciprocating assembly without carburetor, muffler, shroud or anything else. Also, a good mower man can hone the cylinder and put a new piston in. Whether or not this should be done depends on the cost-effectiveness of just replacing the engine vs. repair. A decent home mechanic could make that work, but at a shop you'd be better off putting an engine on it.

The smoke is blue/white, not black. I can also smell the oil when it happens.

Perhaps my choice of wording was poor; the engine wasn't jammed. Instead, it just wouldn't "catch" at all; not even a little bit. One short spritz of carb cleaner and it started on the first pull. (So I suspect a minor carb blockage there...) I haven't had any starting issues since. (The mower has always been a bit reluctant to get going for the first mow of the season.)

Oil is HD30, changed every season, and filled to the correct mark. Fuel is treated w/ STA-BIL at the time of purchase. I'll admit I was somewhat neglectful and didn't put oil in the spark plug whole at the end of the season. However Raleigh isn't particularly humid in the off-season anyway.

I have a can of seafoam lying around somewhere; I'll give it a go. How much should go in the sump and the tank?

I'd probably go the $85, shipped, short-block route before honing the cyl. Once I had the new piston, rings, head gasket, cyl. hone, and quality caliper, I don't think I'd be too far off of $85. (Jack's Small Engines lists it as $50 + shipping just for a +.020" piston/ring kit.)

One side note that is maybe relevant... The first oil change ever for this engine (after the specified 5 hours of use) looked like silver paint. Since the engine ran (and ran fine) for the whole five years of the warranty period, I couldn't file a warranty claim, but I am thinking now that perhaps this was an indication I wasn't exactly going to get 20 years out of this engine. Maybe a new engine would make the most sense after all... I can maybe squeeze a couple more years out of this one before it finally starts bogging down in routine stuff.

"I have a Toro SR 20042 that is burning some amount of oil when mulching thick patches of grass. (I can see faint smoke and smell the oil burning.) I've changed the oil every year, just like I should, so I'm kind of disappointed this is happening after only seven years of average use. (1/3 acre plot, of which the lawn is only a portion.) It is not burning so much that it is appreciably moving down on the dipstick yet, but you certainly can see and smell it. I know I'm in the early stages of any problem since it doesn't really bog down that much even in spots where it is probably too tall to mulch."

After rereading your post a couple of times I'm struck by comments about how the burning smell only occurs when mulching a thick area of grass as well as how it doesn't seem to be too much of a problem even when mulching grass that is too tall to mulch.

I wonder if you are not trying to mulch too much grass at one time. That combined with gas that might be old and an engine that is no longer new might produce a bit of smoke. My suggestion would be to mulch more frequently, mulch half a row or bag if the grass has grown too tall.

In the unlikely event that the engine is really burning some oil I would not put $250.00 into the mower top fix a minor problem like that.