I just did this from Harvey, I didn't do shit with the mortgage company.
Got my check from TWIA(in my name), cashed it, showed my roofer what TWIA gave me, they worked with TWIA and got me another $400 check(in my name) I cashed that.
Roofer did the job under quote so I actually saved about $400 on my deductible. I gave the roofer their check(from my savings account), a couple months later I got all the TWIA paperwork required.
At no point was there anything to do with my mortage company.
Luckily for me, between TWIA and the roofing contractor, they new what was going on so I didn't have to stress too much.