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.