"Casualtyship" bias is just another form of survivorship bias. If you avoid counting the vaccine shots that don't result in injury, you're intentionally collecting bad data.

If you equally invite both injury reports and non-injury reports, the money would still be bribery and would worsen selection bias.

Despite the scare quotes, vaccine injuries are a real thing. There is already a reporting system for them.

The claim (by the anti-vax crowd, or the broader vax-skeptical crowd) is that only a small fraction of injuries are reported, and so we don't have a true picture of the danger. Why are only a small fraction reported? Because, unless the injury happens very quickly after vaccination, it's on the vaccine recipient to report it, and most people won't bother.

But this gets difficult, because you get into "after this, therefore because of this". The further removed in time, the less clear it is that the vaccine is the actual cause. (Was that anaphylactic shock really from the vaccine, or was it from the shellfish?)

Is paying doctors to report these the right answer? I'm a bit skeptical - the incentives are for over-reporting. You could say that at least the bias from this will be in the opposite direction from the current bias. But if you want accurate data, is that actually an improvement?