I don't see any reason to raise here as a general rule but against this fish its a bit more complicated. Fish generally don't bluff rivers, but that definitely doesnt mean he isnt overvaluing a straight/set here and playing it like the nuts (so not bluffing as far as he is concerned)?
44 is really the only set in villains range here given his check back on the turn, even passive fish tend to bet sets. There are a few straights, but not that many. The 35% of his range he folds contains a lot of 3's as a general rule so we are still looking at like A3, 34s, 35s, 33 really. Even this kind of loose fish can usually find a fold with the likes of 73o T3o preflop. I'd give him a general range of 33,44,A3,34,35 that we beat. Theres only really 3 or 4 combs of diamonds that have us beat, KQdd KTdd 9Tdd. I'd expect even this fish to bet Axdd on the turn, although maybe not so we should probably include the 5 combos of Axdd in his range also.
So villans entire range is probably like:
We can maybe even widen it to include some 2P and spaz hands.
Your like 80% or something against this range, so if you feel villain will call a raise with the entire/most of the above range it is a clear shove. If we are raising here i feel it should be a shove, as you say we pretty much have to call off any reshove by villain even if we make it like $13 if we think he plays a straight/set as the nuts, yet we give him the option to just call losing $6-7 in value when we make it that size. Any hand he is calling a raise to $13/14 with he is calling a shove, so go for max value and ship it.
It does suck being this deep sticking it in without the nuts but I think we lose to much value against this particular villain by calling. Although calling is by no means a bad play, just less profitable (albeit lower variance). Just expect to be beat sometimes.