The Constitution of the United States says but one thing about contracts.

Article I, section 10, clause 1:

No State shall enter into any Treaty, Alliance, or Confederation; grant Letters of Marque and Reprisal; coin Money; emit Bills of Credit; make any Thing but gold and silver Coin a Tender in Payment of Debts; pass any Bill of Attainder, expost facto Law, or Law impairing the Obligation of Contracts, or grant any Title of Nobility.

As Wikipedia states:

The framers of the Constitution added this clause due to fear that states would continue a practice that had been widespread under the Articles of Confederation—that of granting “private relief.” Legislatures would pass bills relieving particular persons (predictably, influential persons) of their obligation to pay their debts. It was this phenomenon that also prompted the framers to make bankruptcy law the province of the federal government.

Knowing that we do that the US Constitution is a document of enumerated federal powers we look to Article 1, Section 8, Clause 4: The Congress shall have Power

To establish an uniform Rule of Naturalization, and uniform Laws on the subject of Bankruptcies throughout the United States;

It would seem to me that “a proposal, popular among many Democrats, to allow judges to modify mortgages in bankruptcy court.” would be unconstitutional as singling out mortgage contracts as judicially modifiable would make bankruptcy law non-uniform. I doubt they are currently but that would not legitimize this additional infraction.

It would seem to me that “uniform Laws” applies to the topic not simply their reach. Otherwise the Congress could create special exceptions for whomever they wished so long as they group things narrow enough.

Regardless of what the Constitution says I’m still not OK with an arbiter single handedly changing a contract without explicit consent by all parties of that contract. In this case it would obviously be used to help the mortgagor over the mortgagee. One could claim that by going to the government court you are giving consent but I doubt that completely private bankruptcy arbitration is something which is generally accepted. So I’m not sure that there is much of a choice in reality.