If a physician performs an abortion in Texas, then immediately makes a payment of $10k to the person of their choice for violating SB8, are they immune to further damages under the bill?
Almost certainly the answer to this is no because SB.8 talks of the $10K as being (minimum) statutory damages. (The actual provision 171.208(b)(2) is for damages "not less" than $10K, i.e. the court can award more in the first suit.) If e.g. someone breaks the Fair Debt Collection Practices Act by making spurious claims against a group of people, they are each entitled to up to $1,000 statutory damages each (and class action is specifically allowed/mentioned). Generally, you can't avoid paying damages to an injured party by paying them to someone unrelated (before). There's nothing in SB.8 to suggest otherwise, the wording is:
a court may not award relief under this section in response to a violation of Subsection (a)(1) or (2) if the defendant demonstrates that the defendant previously paid the full amount of statutory damages under Subsection (b)(2) in a previous action for that particular abortion performed
Only proof of payment (not even having lost another suit on the same incident in another county) precludes a judge from awarding the statutory damages. (Claimants can file suits in their county of residence per 171.210, which me not knowing the Texas civil procedure too well, seem to allow multiple parallel lawsuits to proceed.)
Now SB.8 is special in that limits relief to the first who sues (or maybe the first who wins) while enabling a very large class to claim injured status and sue... which in itself doesn't make a lot of sense when speaking of statutory damages, but it's a novel kind of relief, so who knows what courts may decide. (It would be interesting if a court determined that that first-claimant limit in SB.8 is unconstitutional, under either Texas or federal constitutions. In some other state laws, there is sometimes a limit on the total amount of damages out of a single incident, but it's not awarded on a first-winner basis, instead there's a provision to "allocate to each claimant his equitable share of the total".)
Granted a defendant could come with a legal strategy that look something like: when sued in Texas county arrange for a "friendly" organization (members) to immediately sue in another county. Fight the first lawsuit so as to delay judgement, but immediately concede the one from the "friendly" organization members, so that only the "friendlies" get the statutory damages (first), possibly e.g. donating them back to the defendant thereafter. Whether this would work really depends on details in the Texas civil procedure law, which I'm not too familiar with.