"2018 and 2019 AGI's affected the EIP which in turn affected the balance of RRC on the 2020 return."

That's not how to understand it. The most recent info on hand your filing, to the IRS, was used to Project the EIPs. EIP 1 came out early enough that many people had not yet filed their 2019 income tax form, so their 2018 values were used for projection of 2020. Then, the second EIP was enough later, that 2019 might have been filed and it got used for the projection of 2020. For any taxpayer, then, possibilities include that 2018 was all that was available; or, 2019 was all that was available; or, the EIP projections were split. Everything was paid out as a Projection for 2020 qualification.

Both 2018 and 2019 AGI data are meaningless for the 202 tax year. The only Actual value that is used, is the 2020. The payments are reconciled to what the taxpayer is qualified for, and the qualification is based on the 2020 tax form. That is the reconciliation of what they qualify for against what they already got.

Overpayment: There is no provision for Repayment to the IRS if your advanced payments ended up being more than you qualify for based on 2020 AGI.

Reduction: The first EIP can have been offset to cover back child support. That doesn't apply to EIP 2. The refunds, even if created by a refundable RRC (credit not prepaid already) are subject to offset the same as any other refund can be. The offset program states: "TOP will continue to collect all certified delinquent federal nontax and state debts through TOP from all eligible payments. For debts that remain active for collection in TOP, offsets will continue until a debt has been paid in full or until we have been directed by the federal agency or state to stop offsets."

"All BS aside, how else can we prove client did not receive?"

You can't prove the negative, but you can find out how much their were issued, if any. And you might be able to show they got it, and just forgot. By using the IRS access tools for either the taxpayer or the preparer to get the taxpayer's Account Transcript, which will show the amounts. Then, you would need to cross-reference to the taxpayer's banking, or submit to the IRS for a replacement of the lost payment card or to reissue the check. Start with their transcript, to know what you should be chasing or not chasing.

"(because I have their 2018 and 2019 AGI info. Very relevant.)"

Those AGI are moot, now. The payment amounts are relevant. Two completely different things. And for all you know, and we all could do the math, 2018 AGI might have been used for this taxpayer's EIP 1 but not EIP 2. We would still need that info and their filing status, as noted previously.

"Client claimed $1,936 of RRC on Line 30 of Form 1040.

IRS adjusted it to $355. (Thus, Client was eligible to something.)

The difference was $1,581.

By shear coincidence, Client received a total EIP of $1,581 ($1,066 Payment I, $515 Payment II, per Gibraltar-solid info)."

Not by sheer coincidence. It's the way the math works out.

But you seem to be locked into the IRS didn't allow the RRC and that's not how it seems when you present it. Without knowing filing status and AGI, it's impossible to confirm what they qualify for against what they got.

Line 30 was supposed to be any Remaining Credit to them, that was not already in the prepayment amounts. Not the full credit; just the difference if owed to them.