First off, just to be clear: Most of Harvard University is in or near Harvard Square, but there are some outlying parts. (the Business school is in allston, and the medical school is in Boston, for example.) I'm assuming the program is in the Harvard Square area.
I left Boston/Cambridge 35 years ago, so I can't really talk about the area where the Boston Seaport District hotel is. I think that's a somewhat rejuvenated neighborhood but it's a bit of a slog to get to Harvard Square from there. The Doubletree is actually within a long walk of Harvard square: about 1.3 miles according to google maps. So I'm thinking you'd find the Doubletree much more convenient. And with a shuttle, that's even better.
I do see that there's a Trader Joe's not too far from the DoubleTree: it's across the river, and about a 13 minute walk from the Doubletree (1/2 mile according to google maps). By the way, there's a Courtyard Marriott right next to that Trader Joe's, though I'm guessing you already considered it and ruled it out based on price, amenities, availability or something else. Just thought I would mention it.
Staying way over in the Seaport hotel sounds too far to me personally, but this is your trip and your money, so you have to decide based on your own preferences. I guess you could list out the pros and cons of each and make a choice based on that?
Also, I don't know what you are planning to do while your daughter is in the program; are you the type who just likes to stay in their room and read/watch TV, or do you plan to get out and explore the area? The last time I was in Boston was 2019, and we stayed in an apartment on Beacon Hill. I had hurt my achilles tendon and so couldn't walk too far, but I could ride a bike with no issues; Boston had started a bike sharing program called "Blue Bikes" and I signed up for a membership (which I cancelled after my trip) and this allowed us to rent bikes for transportation. You can only keep the bikes for a relatively short amount of time (perhaps 2 hours?) but it was absolutely great for getting around. We rode from Beacon Hill to Harvard Square, etc. Very convenient, and there were lots of bikes available.
I don't know if the Blue Bikes program is still going on and if it's as good (in terms of bike availability, etc.) but it's something I wanted to mention.
One more thing: Boston in July can be quite humid, so you might want to factor that in to your plans.
And I don't know if your diabetes lets you eat ice cream, but Boston/Cambridge historically has great ice cream shops.
I hope you have a great trip!