I use a reciprocating saw. But, I know how to be careful that I don't either cut the floor joists or the sub-flooring. The key to doing this job, no matter how you do it, is to get out the 'bad wood' as close to the wall as possible. If you have to use a hammer and chisel, that's what you do. If you're using a circular saw, just be sure it's set to the thickness of the wood! Otherwise, you may have more problems than you want.
If you're having a pro install the new floor, see what it would cost for him to do this part of the job. I'm sure you saved some $$$ by doing the major demolition work. Or, rent the saw and have a competent 'friend' do the remaining cutting. Others???
My best to ya and hope this helps.
PS: Try to use screws for your new underlayment and be sure they're recessed, and the 'holes' are leveled! The same goes for inbetween the sheets. Also, make sure your underlayment runs perpendicular to whatever you're installing it over top of.