Which is better, a cash-out refinance or a second mortgage?
If you can get a traditional 2nd mortgage from a lender with reasonable rates, that's what I would do. The closing costs should be much less that way; especially if the rate you already have on the first mortgage is pretty reasonable.

We filed chapter 7 bankruptcy in 2010. Our bankruptcy paperwork shows that we did not reaffirm the mortgage. However, I just
Sounds like there are some credit reporting inaccuracies. I would talk to another (new) attorney that specializes in bankruptcy law.