I agree with the other poster that you need to define what you even mean when you say free will. IMO, strict determinism is not incompatible with free will. It only provides the mechanism. I posted this in another thread where this came up:

The implications of quantum mechanics just reframes what it means to not have free will.

In classical physics, given the exact same setup you make the exact same choice every time.

In Quantum mechanics, given the same exact setup, you make the same choice some percentage of the time.

One is you being an automaton while the other is you being a flipped coin. Neither of those really feel like free will.

Except.

We are looking at this through an implied assumption that the brain is some mechanism, separate from "us", which we are forced to think "through". That the mechanisms of the brain are somehow distorting or restricting what the underlying self can do.

But there is no deeper "self". We are the brain. We are the chemical cascade bouncing around through the neurons. We are the kinetic billiard balls of classical physics and the probability curves of quantum mechanics. It doesn't matter if the universe is deterministic and we would always have the same response to the same input or if it's statistical and we just have a baked "likelihood" of that response.

The way we respond or the biases that inform that likelihood is still us making a choice, because we are that underlying mechanism. Whether it's deterministic or not it's just an implementation detail of free will, not a counterargument.

Now here’s an interesting take.

In principle, one might be able to predict behavior based on this model.

But I would asset that it is not possible to achieve these conditions no matter what godlike technology one has.

Let’s go simpler. We don’t want to predict a human we want to predict the path of one electron.

Starting from initial conditions we should be able to predict the path of that electron right Wrong!

It’s wrong because it is impossible, in a way that cannot be overcome in this or any universe, to know those initial conditions.

And that may seem like a technicality, but that’s exactly where the chink in the armor is: no matter how precise your model, it’s impossible to determine the state of a closed system, because it’s closed, and it’s impossible to predict the behavior of an open system, because its evolution is determined by its interactions with its surroundings, and you can’t get all that information.

So the idea of using physics to predict things precisely is a Platonic ideal, not a thing which can manifest in reality.

The brain as far as we know works like nowadays "ai" it makes assumptions and tries to get the best and most fitting outcome for itself, this decision is influenced by a astronomical amount of data (thats the difference to nowadays ai) and (for the lack of better scientific measurements) emotions, meaning that theoretically it is possible to perfectly predict your choices, however, if you know about said prediction it will influence the system again.

Saying you have to pick between a Red and a Green Gummy Bear with the exact same taste, you will probably pick the red one because the subconscious associates red usually with food (red and yellow, that's btw why many fast food chains use those colors) however, if you are aware of this, this will likely end up in picking the green one as rebellious act against nature, wich again is predictable because humans are self aware and don't want "to be slave to their own instincts" now if you know about the prediction, the only way to "be unpredictable" would be to not chose a Gummy Bear thus "breaking" the test... Wich is predictable. This is of course a very simplistic experiment but gets the point across, you can't be unpredictable unless you break the laws of physics. This does however not prove or disprove the "free will" inside the system you are in, you could chose things freely, you just don't because its unnecessary and time consuming.

TLDR: we don't know.

I believe your scenario conflicts with Heisenberg's uncertainty principle. As far as I know it's one of the most ironclad laws of physics. As a quote I found in Wikipedia puts it,

The uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.