I could get by for awhile on my prior salary, but it would be hard and
not sustainable long term.
Should I offer to delay the raise?
Don't do it.
If the company is willing to share the burden and make an across the board salary cut (except for the lowest-paid employees), or make an offer of unpaid time off, then maybe, consider doing one of those things. But otherwise, don't. Don't unilaterally offer to take on that burden all by yourself.
After all, chances are that you've been underpaid for many months if not for many years. Also, since there is no transparency as to how much others are making, or how much the company is making, it's clear to me you're not being viewed as a real partner in such decisions.
Also, while you showed wisdom (or luck) in remaining with your current employer when so many other new employers are currently rescinding contract offers or delaying starting dates indefinitely, accepting the counteroffer from a current employer was also a risk.
The fact is some employers purposefully make a counteroffer, only to retain an employee a little longer, so they can fire that employee on their own timetable. Now, I'm not saying this is common, nor am I saying this is going to happen to you. But if this did happen to you, you would be much worse off with no raise and no competing offer during this time of crisis.
And if not for your self-interest, think of your sanity. If you forego your raise today, and tomorrow, your employer orders a $10,000 foosball machine, it will infuriate you to no end. And it doesn't have to be a foosball machine, it could be your company blowing the money on brand new furniture, or the owner buying a brand new BMW. My point is every time money gets spent frivolously, you will think of it as your money being spent. And this can't be good either.
Again, if a sacrifice must be made, everyone should share in the sacrifice. and everyone must be on the same page. Otherwise, you will come to regret your individual contribution. Making a shared sacrifice also ensures that the sacrifice doesn't last as long, has more of a total impact, and is less likely to be reneged on when it's finally time to undo.