If you run
pkg> update
pkg> rm FFTW
pkg> add FFTW@1.1.0
What do you get? I think (though I’m not 100% sure) that this should show what is holding FFTW back. Note that if the command works then it will pin FFTW at version 1.1.0 forever, which you probably don’t want long-term, so you should run pkg> free FFTW afterwards to un-pin the version.
The other guidance here is that as you’ve discovered Julia has per-project environment support built-in. It’s good practice to have a separate environment for each thing you’re working on, so you don’t end up with a big cluster of packages, which makes this sort of thing more of a problem. That said, I realize it’s not very satisfying to run into this issue and then not have a clear path to fixing it.