This doesn't sound like a very good idea unless you have equity in your property. If what you want to do is reduce your interest rate, the trick is to modify your first mortgage to reduce the interest rate and settle the 2nd mortgage for a lump sum buy out. If you are patient, and if the property is underwater, the 2nd mortgage may be sold to a debt collection company if you don't pay. Once the debt is with a debt collector, the collectors may take a cash buy out of the loan for pennies on the dollar. I typically see 10 cents being a settlement, or $5,00o on a $50,000 loan, but only if there is no equity in the property. Hope this perspective helps!
First things first. As a result of the chapter 7 discharge you have a first mortgage on which you remain personally liable (signed reaffirmation agreement) and a second mortgage on which you PERSONALLY do not owe any money on.
Nonetheless, both the holder of the first mortgage and the holder of the second mortgage retain valid liens on your property.
There is no prohibition against refinancing for someone in your situation. However, if you are wanting to refinance through your second mortgage holder, that will not work. They will contend that any effort for them to refinance the second mortgage (even though combined with the first mortgage) violates the bankruptcy discharge injunction -- and they would be correct.
As for pros and cons, do the math. It is no different than any other decision to refinance. Note that through all of this I am assuming you have more that enough equity in the property even after paying off the first and second mortgages.
The issue isn't really "can" you refinance, but "will" a bank do so.
Your chances of refinancing if your loans are with two different lenders are in the range of the proverbial ice cube--no sane lender will advance money to get an upside down mortgage.
Sometimes, though, if both are with the same lender, you can refinance with that lender.
Also, you may be able to modify your first mortgage. And, if your first is upside down, you may be able either to file a Chapter 13 now (a "Chapter 20) or move to revoke your discharge and convert your case to Chapter 13 to strip the second mortgage off.
You should really consult a local Chapter 13 attorney, as this varies wildly across the country.