TL;DR: If I am untypical, then many more observers exist. If this claim is necessarily true, then classical SIA is false.
There is a situation where we could prove something like SIA, let’s call it Untypical SIA. Imagine that there are 100 green dots and one red dot on a bow’s target (and they cover all the space). We know that there were either 10 or 100 attempts to hit the target with a random arrow. We only know that the red dot was eventually hit (maybe we are in it). Now we can give higher credence to the theory that there were 100 attempts (P1(red)=0.6339) over the theory of only 10 attempts (P2(red)=0.0956) to hit the red dot. Credence in 100 attempts = P1/(P1+P2) = 0.86.
The resonating above is made from God’s view and seems to have high epistemic status: it is not an assumption, it is a proof.
Examples of application of such untypical-SIA reasoning in the real life: if there is life on Earth, there are many other planets out there, which tried but fail to develop life. If humans are the only intelligent species, there are many others species that “tried” and fail to become intelligent. If our universe is fine-tuned, there are many other universes which are not fine-tuned, so the multiverse theory is true.
If Untypical SIA is a necessary condition for a larger set of observers, then classical SIA is false.
Untypical SIA requires that the red dot is different from the green dots. If all dots are red, and we learn that one of the dots was hit, it doesn’t give us any new information to update. If uniqueness is the necessary (=if and only if) condition for update when the lack of uniqueness means no update. In that case, classical SIA is false (it is not a problem in the infinite universe, where something like SIA still works, as we should look at relative concentrations of observers in different regions of the multiverse instead; all our SIA-based cosmological constructions are safe herein).
In other words, the fact of my existence is an argument in favor that more observers exist – only if I am special. In classical SIA, the mere fact that I exist at all is evidence that many more attempts to create me were made, without any mention that I should be different from all other observers.
Obviously, I can’t just say that my random name is rare. Because anyone can say that, and we are all in the same “epistemic situation”. If it is an experiment, un-typicality should be defined a priori, before the experiment.
I should count as unique only something which a priory can be considered unique, like a name AAAA AAAAAA or a time of my birth like 11.111.1111. Having consciousness or being fine-tuned are examples of such a priory uniqueness, but being named “bgjkuytvnki” is not.