A photo would tell a lot. Option 1 is the only way to go.

Agree with Seabornman. 2” XPS glued to the block and seams taped. You get what you pay for. XPS will not absorb moisture. Don’t forget fireblocking.

I understand option 1 is "100%" but if option 3 is "98%: for hundreds of dollars less", I'll take option 3. What are the risks?

Also there is a french drain opening around the perimeter and I don't want to seal that over with foam

Why are you opposed to taking down the framing and putting it closer to the wall? You could reuse the wood (not that wood is expensive).

Saving 6" of space all around the perimeter is significant and it will make adding the XPS to the walls so much easier without the framing in the way.

Options 2 and 3 will lead to mold, insulation saturated with water, and rotted wood eventually.

Why is that? When there was no insulation at all there was no moisture.

Simple explanation, you had no vapor barrier with 8” of space and hardboard paneling. Dampness moved into the room. I’ll bet the seasonal humidity in your basement was very high. Perhaps water pipes showed condensation. Since you need French Drains you have a moisturizer problem - it doesn’t just go away.

Humidity ought to be maintained below 45% using a dehumidifier if necessary. Wood, drywall, carpet will deteriorate if humidity isn’t maintained. The bottom plate of your stud wall should be PT atop a foam strip. We’re only sharing knowledge with you to make your basement comfortable and lasting.

No air gap; air gaps will form a thermal convection loop and short circuit the thermal design. Low perm foam against the block is fine, past that no vapor barriers or low perm materials. My typical basement wall is xps/extruded polystyrene, stud framing w/unfaced fiberglass, then drywall and high perm paint. The foam/fiberglass interface temp is raised by the foam layer so condensation within the wall won't occur and any water vapor will dry towards the interior as roberts pointed out.

Read all about basement walls at https://www.buildingscience.com/

I cannot avoid the air gap, I need to work within that constraint. I know when people are online they love to swear by the optimal solution, but its not realistic when you factor in the cost and time. Yes if I had unlimited time and unlimited budget I would just demo the house and build a brand new one.

My question is if I used faced batt insulation on the studs and maintain the air gap between the unfaced side of the batt and the concrete wall, will there be condensation on the insulation, if if so why. I would not think enough warm humid air could get through paint, drywall, facing, and insulation to eventually make contact with the cooler concrete wall.

Please strive to be helpful and address my questions in the terms of the constraints I listed, not "Forget everything and spend any thing it takes to do it the 100% best way possible". Not everyone can afford a Porche, I'm just looking for a Kia.

The face on batt insulation is not an adequate vapor barrier. If you could drape 6 mil polyethylene on the back of the studs, fully sealed to the floor and somewhere above, and were able to tape the seams, that might provide the vapor barrier required here. I don't know how to do that.

Moisture sensitive insulation in the stud wall with a gap between it and the foundation will result in: 1) convection loops that will drive moisture into the interior framing, promoting rot and 2) moisture in the fibre glass from the inward drive of water vapour from the exterior.

Don't take the word of any poster here. See the references from Building Science Corp., headed by the leading Phd. building scientists in North America.

BTW, the video "expert" misses that, among its other functions, housewrap such as Tyvek, lets moisture move outward through the wrap. So that means the moisture in the space between the Tyvek and the foundation moves through the Tyvek into the fg, feeding mould growth.

From Jon Eakes linked article:

It's not a question of a Porsche vs a Kia, it's a question of what works and will last the test of time. If XPS foam is too expensive, go with EPS foam. If that's too much, fill the gap and stud bays with unfaced fiberglass, completely, no air gap. Then drywall and high perm paint. And hope for the best.

Fibre glass in a basement wall. Omit the poly and get the same effect with high-perm paint. Source: Building Science Corp.

^ That basement has fiberglass in direct contact with a basement wall. At no point did I suggest doing that.

The basement has a french drain installed and there is a 1 inch gap between the floor and the exterior wall. If I went with a rigid board it would cover that gap. Is that OK?

If I used rigid foam against the walls, would It just be Wall > Foam > Air > Drywall? Nothing else just board on the wall and drywall on the stud?

Here is what the walls and framing look like. The black crap on the floor is some old adhesive, its not mold. You can see the gap for the French Drain.

I just want the cheapest way to insulate this wall that won't grow mold or mildew and will prevent the basement from smelling musty. I don't care at all about thermal performance. The floors will be vinyl plank flooring with the built in underlayment.

Foam against the wall, whether it's rigid or sprayed, is the the most effective system.

Here's how Building Science Corp. advises doing an interior drain. Many drainage companies use this method.

This way, interior insulation can be taken directly to the floor. Glued, but not nailed/screwed.

For a basement with a system that requires a gap, the best you can do is try to prevent the damage of the convection loop with moisture resistant insulation at the top--rim joists and floor members. The convection loop problem was only fully realized in this century. Before then, in fact, it was often advised that half-height basement insulation was fine, as it was the above-grade portion of the basement wall that needed to be upgraded.

It's hard to picture your situation in the abstract.

Yes, wall, foam air, drywall is safest route. I have added unfaced fiberglass batts in the studs with that assembly in a "dry" basement with success, though some would tell you not to. I would keep the foam 1/2" above floor to allow for drainage.

Wouldn't leaving a 1/2 inch gape create this dreaded "convection loop"?

I'm no confident that JUST the foam will stop any cool air from hitting the warm drywall and condensing right on it

There is very little movement in the space between the insulation and the the drywall. (The drywall with latex paint is the vapour barrier.) And I'm projecting that the 1/2" gap will not contribute much to creating that loop. As well, there will not be much of a temperature differential between the bottom of the foundation wall and the room temp. To be really sure to limit damage, you can use water resistant drywall.

Here is the relevant explanation from the Eakes article linked earlier.

Note: For the foam to be effective, it must provide at least the RValue prescribed by IECC 2018 for your climate zone.

I missed this pic.

Now I see your dilemma. Instead of using a drainage mat, the drain simply picks up water running down the foundation wall, so you can't block it with insulation.

However, the original post says, "[t]here was previous dampness penetration in the walls, but those have been remediated."

If that's the case and the walls are dry, you can mount the foam with no problem.