In a complicated situation for 2017 taxes (settled in 2018) (Estimated State payment too big --> money back; + 2018 Tax Reform), it seems like I'd be better off had I made a State Tax estimated payment in 2017, instead of making it in Apr 2018.
Is it possible to amend my 2017 tax return to "move back" that payment, even though I guess it would mean I'd owe interest on that amount from 31 Dec 2017 to Apr 2018?
No, amending a return allows you to correct misstated facts, or choose an alternate permitted tax treatment consistent with the facts, but not to calculate as if something happened in a way other than it did. There does not appear to be any provision allowing you to retroactively treat such a payment as being made in a different year.
Definitely don't. Nanoman told you one reason. Here's another.
As far as IRS is concerned, OP would be creating "churn" unnecessarily.
By January 2018, it was well known that 2018 taxes would be changed. It was known most tax brackets would see their tax drop by 3%. So if someone could take a tax deduction in 2017, and then pay income tax on that same amount in 2018, they would be 3% richer -- or in OP's case, $600 richer on $20,000. There aren't many opportunities to do that in the tax code, but by golly, State tax prepayment, deduction, refund, declaration is one of them. I'm not saying OP planned this in 2018...
... but it looks like intentional churning for that purpose. Especially now that the 2017 taxes are settled.
I stick by my earlier advice: Don't take the $20,000 as a deduction in 2017, and don't declare it as income in 2018 (i.e. exclude that amount from the "state tax refund paid" section on your 2018 1040). You did not benefit from the deduction, so there is no question you do not need to pay tax on the $20,000 of the refund.
How exactly you claim that appears to be complicated because your overall state payments were made in 2 years (man, I'll score a mental note never to do that.) If you already have a tax accountant, let him figure it out. IRS does not penalize you for mistakes made by your preparer.
