I think the winner for smallest size of ship for a 76 mm gun has got to be the Italian Sparviero class:
However, you have to keep in mind that the reason these small ships can have a 76 mm gun (or put it in a sea can) is that they are only composed of the gun and its handling system (on the deck immediately below). It also means that they are restricted to what's in the ready use automatic loader. No magazine to reload from in action.
That may be fine when you are on a small local patrol vessel and can go back in harbour to reload after any encounter, but not so good on a frigate/destroyer doing mid-ocean escort work. The Flyvefisken with the sea can could not reload. The Iver Huitfled have a 76 mm magazine, but the reloading from the elevator to the loader inside the sea can has to be done by hand from the service door, and its a cramped can to work in.
Anyway, it would have been just as easy to put a 76 mm on the HAL as a 57 mm. Just as we could have kept the 5 inch Otto Melara on the IRO instead of replacing them with the 76 mm. The choice is not based on design constraints but on operational use.
Modern warships fight one another at long range with missiles these days, not at short range with guns. So the guns serve one of three basic purpose: fighting asymmetric threats, Anti Air warfare or ground support. The larger guns, from the French 100 mm up in size are considered primarily ground support guns. The smaller guns of 57 mm and lesser calibers are primarily for asymmetric/AA work. And the 76 mm is a weird animal that is neither fish nor fowl right in the middle. It does it all, but none of them as well as the other smaller or larger "dedicated" calibers.
So the choice of the 57 mm for the HAL's was not a design decision (as in: it's the biggest we can carry) but an operational decision: what is the most likely threat that this class of ship will encounter that require a gun: lets maximize that use. AA/asymmetric won and we got the 57 mm.
The choice is not that bad when you think of it. The last time the RCN did ground support with guns was during the Korean War to do train busting. It required the destroyers to rush inshore, and thus expose themselves to battery fire from the shore, so they could fire from inside the gun's range. Nowadays, with land attack missiles, why would you want to do that? And otherwise, what are the chances of Canada needing to carry out ground support with guns of Canadian troops? Pretty damn near zero. So, good choice, people!