Hi,
Just thought I would put out a feeling or thought about your situation. I'm sure I have no quick fix answer but I'm bored again at my winter house and sorry but here goes!

I have one K-Jet car. It's a '78 stick shift model with a Lambda Sond system.
You have brought up a good question of how does the engine adjust for idle load of an automatic transmission. I have no answers for that one. I have never seen both together or driven one.

I do know that on some cars there is an electrically controlled vacuum port valve that adjusts the idle up for an air conditioner. These models came around about that time as A/C became more popular though the years.
They allow more air in to kick up the idle electrically. Maybe your are missing that feature for a auto trans and it is not working. Does you car have an A/C compressor on board?

On the CIS, I have, the idle is already at 1000 rpm as standard set speed with the throttle plate screw and mixture knob on the throttle body after the car is warmed up.
This is done in conjunction with the auxiliary air valve that you say you have changed out several times.
The AUX valve works independently from the rest of the car. It has an internal spring tension held by a pin and nut on the outside of the valve. The bimetallic hook on the other end slackens or tightens(?) the set tension when heated.
It holds the idle up higher than 1000 rpm when cold and is heated electrically "to mimic" the engine warmth to lower the speed as it warms itself up.

The engine and AUX valve have to be timed to warm up together. Warmth affects speed of the engine as it overcomes engine cold friction and the incoming air of which affects fuel density as it goes directly into the intake manifold.
The AUX valve is mounted on top of the engine to sense both environments to be an average.
The cold RPM setting is not published for a reason. I theorize it is set by a sampling for an average of the number of "your own" normal starting days. I shoot for about 1200 rpm on days in the low end of the 50's.
It can go as high as you need it, cold, but should it's not be adjusted so high to override the base line throttle settings when it gets hot from its own heater.

The AUX valve closes down the bypass air going around the throttle plate but all the air should still come in under the flap. In itself, that it can be a hunt to eliminate on any car.
The AUX valve is a static air idle control, where on the LH, the IAC is more dynamic to cover both areas all at once. The computer controls a lower 750 RPM idle for less emissions and economy.
I think, another theory, the extra 250 rpms are needed on the CIS to get the air flow to be more stable or responsive passing under the flap.

The same goes for the heater in the control pressure regulator. It's used in the fuel controlling devices down below the intake manifold.
It has heaters that will adjust fuel pressure to the fuel distributors flap swing height. Mixtures are in relation to air flow amount right there.
These are independent of the AUX air valve as mention but are "mimicking again" the engine warm up.
I guess you could say, it is about balancing the averaging!

This is where gauges will have to be used to see what's going on in relation to each other, down below, during and after a warmup.
This fuel pressure adjustment system is like a mechanical AMM of the LH system.
The AMM replaced the fuel distributior measuring device part.
Only with th CIS system, with the gauges, you can see this mechanical mixture control pressure function working mostly during a warmup and overall functions that are preset AND have specifications. The AMM's are under a remove and replace specification, of which, you see here all the time.

When set correctly, they stay that way for a long time, provided, "fresh & clean" fuel is ALWAYS kept in the car! 3 months or more and our gas is starting to go down hill nowadays!
The CIS is very reliable in a sense of mechanical versus electronic failures seen in later years.
Being gummy gets both systems but heat and corrosion hurts the electronics and is an add on!

This brings to thinking here.
When you open the throttle there appears to be a large time delay or response from the basic mixture needed versus what is needed for the loading of the transmission.
It Could be control pressure issue or just a sticky/dirty fuel distributor. You said the car sat, so if it's old fuel gum in the inlet screens the "flow reserve" maybe too low.
It runs "nicely forever" until you drop it in gear. It Appears to me, it's some sort of general term of "gummy" of operation going on.

I'm totally guessing here because I'm working on your cause and affect statement.

I will say I think that you are all over these are in the right areas.
The TPS should show a "Change of State" with an ohm meter between the terminals when testing. I say this because that's all it does. It's there to be a ECU monitor FOR the plate shaft openings. Going off idle and back on. Closed or open the contacts.

The O2 sensor does control the frequency valve through the ECU to adjust mixture in conjunction with the flap mixture movement. Its a minor thing in some respects but it is part of the whole. Basically, if it's not BUZZING, something is wrong.
I have found that there is a square relay on the drivers fender that works between the ECU and the O2 sensor to get power and information moving. Listen for the buzz if the engine is turning.

That is about all I can help. Air balancing was your first thought and was a good one for idle problems on CIS and fuel for performance.

For ALL I know, the transmission is somehow connected to the ECU. Maybe someone out there knows information on that one?

Good luck and keep us posted. I like mysteries solved.

Phil