I found the answer to my own question - in case anyone else needs it

If the entity overpaid the tax, the overpayment will be applied to other liabilities or refunded to the entity after a tax return is filed.

No entities are able to carry forward an overpayment and designate it specifically or solely to the June 15 prepayment or PTE elective tax for future years. PTE elective tax paid can be carried forward and applied to other tax liabilities, with the excess refunded to the taxpayer. The 565 partnership return does not allow an overpayment to be applied to the following taxable year because these entities’ liability is typically limited to the $800 minimum tax, and these entities do not have other liabilities to apply overpayments to. For these entities, overpayments of PTE elective tax will be refunded to the entity.

For all entities, if the entity overpaid the PTE elective tax, the overpayment will be applied to other liabilities (if any) or refunded to the entity after a tax return is filed.