I think what's bothering me here is that (in my experience, anyway) "engagement parties" aren't given by the couple themselves, they are given by others for them. So if the invitation comes from the HC themselves, there is a real risk that it will be seen as, if not a gift grab, then sort of wanting to have your cake and eat it, too.
That may depend on where (general) you live. Here in Australia it's the norm for the couple and/or their guardians to pay for/host the party. I certainly couldn't imagine expecting a friend or more distant relative to pay for it. That seems extremely entitled.
I guess it really does -- here in the U.S., I've never heard of a couple giving their own engagement party. It's usually something like a party given by the parents of the groom where the wedding will be in the bride's parents' city too far away for the groom's family friends to travel to conveniently, or where the wedding will be very small, and most often when the parents' friends haven't had a chance to meet their child's fiance(s) - like if they live in a different city. Often it is given by close friends of the parents or by aunts and uncles, as one might give a shower.
Actually, the invitations don't usually call it an "engagement party"; more likely something like "cocktail party in honor of Thusnelda and Cuthbert" or "Brunch for Thusnelda and Cuthbert" -- but we end up referring to it as an "engagement party" anyway just for want of a better term for a pre-wedding party that isn't a shower. (Just like, come to think of it, in our community, we call the casual party for all the out of town guests the night before the wedding a "rehearsal dinner" even though it usually has nothing to do with a rehearsal, which probably wasn't even on the same day. But we don't know what else to call it, so there we are.)
That is very much a USA custom. In Australia throwing your own party (be it birthday, engagement, or sometimes even bridal/baby showers) is not unusual or considered gauche.
Rehearsal dinners are also not really a thing. Any additional dinners etc with out of town guests are usually private and unrelated to the wedding itself, and not expected.
I have on a couple of occasions been invited to an engagement (including hens etc) and not the wedding. I find it odd, and sometimes awkward - especially during that period of not knowing if you're not invited or if your invitation is just late. I would definitely go for a later reception/celebration party than a larger engagement. And I would never not take a gift to an engagement party - regardless of the invitation instructions. I might take a smaller gift though.