Welp. It's April 14 today.
I wrote to the school asking if it'd be possible to put into writing multiple years of funding to match another offer. That was 4.5 days ago. I've gotten no response as of yet and since we're 10 business hours (if you count Saturday as a business day) away from end of day on April 15, I guess that's that and that I'm turning down the offer entirely.
I wish they'd respond, at least, but I feel confident that I will be better supported in the long term at another school.
Again, thanks to you all here, especially @TakeruK, @AP, and @rising_star; your thoughtful responses are what I needed to think this through in agonizing detail.