If LSF starts a new process for each job (and only in this case $$ will actually help you) it's not an issue at all, because perl variables aren't shared across processes. In this case you can even use bare word file handles, although I don't recommend them.

Even if it uses perl threads, there's no need to worry because variables aren't shared by default.

(As a side node when you think of a variable variable name, use a hash instead. So instead of the non-working $fh_$$ you'd use my %handles; $handles{$$} = ... instead. But as said above, no need here).