I settled both my father's estate in 1995 (everything went to my mother) and my mother's estate in 2015-16 (everything went to the children). Two of my brothers lived in the house until it had to be sold - it was the primary assets of the estate so I have some understanding of what you are going through.
As far as any specific HAMP modifications, that program was slated to expire at the end of 2017. Congress extended the program through 2018, but basically, that is NOT a longterm advantage because it's ending. As executor and heir (only?) you have two choices, sell the house and pocket the equity as an heir (after going through probate) or take the bank up on a refinance where your name is on the mortgage and deed. If the bank is getting paid they are not going to force a sale, but the probate process will since all assets and liabilities need to be "discharged".
Being on the deed means you are the owner of the property, NOT being on the mortgage means the estate needs to settle debts against assets. If the house was paid for title/deed would transfer to the surviving owner but an open mortgage complicates that process.
My parents home was only in my fathers' name but he willed all his assets to my mother. When I had to settle her estate, we had to get a court-approved modification reopening my fathers' estate to transfer the deed as inherited property to my mother even though she was now deceased, before we had clear title to settle the sale of the house during probate.
It's completely up to the bank as to what terms they allow the mortgage to be assumed - sounds to me like they are being reasonable. You do have an advantage if the real estate has negative equity - you can walk away without it affecting your credit because you are not on the mortgage note.
In my opinion, the original HAMP modification is to a loan that you don't have an obligation to pay nor is it something you should worry about. I do not think you are entitled to any original loan modification (you are NOT on the mortgage that was modified).
If the property has negative equity, or in other words, will sell for less than the mortgage balance you are in a great position as far as talking to the bank. As an example, an appraisal that says the house in its current condition is worth $150,000 and the mortgage balance is $185,000 as the executor you can tell the bank take it, it's all yours (kind of like a quit claim) OR as the "owner" I'll buy it from you for $140,000 with no agent commissions, or advertising costs to the bank providing you finance it at market rates. The bank will squirm and refuse but it's the best solution for them and they'll make a deal - think of it as a probate modification. The bank will treat it as a short sale, but it won't ding your credit since you are under no obligation to pay a penny. You can try this tactic even if there is a small amount of equity since there are thousands of dollars in costs for listing/selling/advertising/recording/ carrying a defaulted (quit claim, short sale or foreclosure) mortgage.
Your attorney should take care of all of this, it's a PITA but it's a mostly logical process of closing out assets and debt with the balance going to the heirs (and a bit to the attorney).
