Just so I am understanding everything you've tried I want to repeat it back:
You're original Speedster with 26 X 1 3/8", S5 rims - the rims are worn and you recently acquired a ladies bike with the same size 26 X 1 3/8", S5 rims (I am assuming a breeze).
You had some older Carlisle tires in 26 X 1 3/8" S5/S6 you had stored and attempted to place those on the ladies wheels. They are tight and do not seat on the ladies wheels but fit perfect on the original wheels.
You then tried the worn tires that were on the original wheels on the ladies wheels and they were also tight and did not seat correctly.
You then went to your LBS and purchase a Kenda tire marked as 26 X 1 3/8 X 1 1/4, (ISO 37-590), but are marked as Schwinn S6 compatible. they are tighter on the ladies wheels and do not seat.
Then you measured the ladies wheel circumference and it measures 2 extra mm more than your original wheels.
Do I have all this correct?
Well I think you have found your problem if the ladies wheels have more circumference (material) than the original speedsters wheels. My only thought would be to get an actual correct new Kenda tire like my picture and try it (marked as 26 X 1 3/8" (ISO 37-597)) as I found that tire did not fit as tight as others. It might give you the room you need to fit onto the ladies wheels. Otherwise you might have to keep looking for some more wheels.
Out of curiosity, what were the ladies Uniroyal Chain tread tires marked as? Can you read them?