But it's possible to ask, what can we change if we can change brain, conditional on the fixed IQ score. But then the "free will" intuition collapses; it's hard to imagine what we could change if our thought processes were restricted in some weird way.

Most people realize they don't have the brainpower to be the next Einstein. On the contrary, the general population has a phobia of anything involving math. I don't think this imposes much of an obstacle against free will. If you restrict to the "It's not exactly rocket science" domain, there are still plenty of choices to be made.

It's true that one can measure his IQ, and that IQ rarely changes much, but still: if you consider IQ fixed and external factor out of your control, then you must consider your thought processes restricted to some set and therefore, not totally under your control.

There are plenty of things restricting my thought processes. There are memory and speed limitations. There's my upbringing and cultural norms. I know my thinking to be prone to error. Free will isn't about being completely free of all constraints. Our courts already apply a "reasonable person" test where they examine what a rational person with ordinary intellectual capabilities could be expected to do. Limits to one's mental abilities have always been accepted along with free will. In fact, most Christian free-will apologists would consider it outright blasphemous to suggest perfect reasoning ability.