I have some disappointing news. At the ski rental counter, you will likely be helped by a teenager who will (it is hoped) measure your foot and pull a pair of boots bigger than your foot. There is unlikely to be a choice of boots (other than sizes). Even if you know to get a smaller boot, it won't be fitted properly to your foot, unless your foot is pretty generic, and it may hurt in spots. Rental boots are designed by their manufacturers to be generic, fairly soft in flex, and pretty generous in size, because they don't make dozens of shapes, and because most renters are beginners and want a boot that's comfortable. If they have more than one type of boot, by all means try them on and see if you can find a snug but non-painful fit.Are people at the mountain likely to help with the fit of rental boots as well as people at my local ski shop? You have a point regarding the possibility of any fit or technical issues while I'm up there. However, if I pick them at a local shop the day before, then I don't have to deal with that in the morning when I first get to the lodge.
I've never rented equipment at Belleayre or Mt. Everest, but I'm generalizing from my experiences in a ski shop. At least our daily rental equipment is better than the season rental equipment.
Renting at your local shop (especially if you can the night before) makes it much more convenient. Inspect the boots and skis you rent so you don't discover something broken or dysfunctional when you get to the mountain. The skis should be fine, although stuff does happen occasionally. Get boots as snug as you can stand without pain. They are unlikely to be really comfortable unless they are too big.
Wear very thin, knee-high ski socks. Don't tuck a baselayer (or anything else) into the boot - socks and feet go into the boot, nothing else. BRING the socks when you try on boots.
Sorry to be the bearer of not-so-great news. And for being so bossy.
The purpose of the rental equipment is to get you on the hill so that you can ski the rest of this season. It's not going to fit really well or be the perfect choice.