Welcome to the list, Marcus. I think your analysis is very good. For some predictions there might be a fixed point; for example, I can predict that I will not commit suicide in the next 5 minutes. Even knowing that prediction I will not try to contradict it. For other things there might not be a fixed point; for example whether I will order chicken or fish at the restaurant tonight. Knowing a supposed prediction I might choose to do the opposite.
Another point is illustrated by your example of using iteration to find fixed points. That is that there are more ways of predicting the future than "brute force" crunching a particle-level simulation. In physics we can make many useful predictions without actually calculating things down to the particle level. For example there are conservation laws that can be used to put sharp constraints on possible future states of a system. It is possible that analogous laws in a deterministic universe might allow for predictions of some aspects of future states of a system without having to go through and calculate the system at a microscopic level of detail. This avoids the problem of infinite recursion since we are using higher level laws to make predictions. So I don't think the argument against predictability based on infinite recursion is successful. There are other ways of making predictions which avoid infinite recursion. If we want to argue against predictability it should be on other grounds. Hal Finney