First of all, I did not find a way to do the reverse: put it all on PA and none on IL. That is, I didn't see a way to adjust the IL numbers in TurboTax, although I guess you could look to see if you could make an income adjustment in Illinois in a similar way as for PA (I didn't look for this possibility).
Second, you said "I was only eligible to contribute to HSA for first 3 months of 2020. All of those 3 months, I was a resident of IL. " This means that you very likely did contribute to the HSA (personally) during your residence in Illinois. This means that the personal HSA contribution belongs to Illinois, not Pennsylvania.
I was happy to see that I could suggest the most reasonable course to you with a method that would work with the existing TurboTax.
I think it would be difficult to explain to the PA authorities why you should be able to deduct your personal HSA contribution in PA when you weren't a resident when you made the contribution.
Make sense?
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