Problem w off piste lodging in Vail is the inability to store gear. There are no storage lockers in Lionshead and very few in Vail Village. Those in Vail Village aren't large enough to store skis or boots unless you get lucky and find a full locker than isn't used which is rare. If you don't stay on piste, make sure the rental company allows you to store gear at the shop. Otherwise, you'll be shlepping gear all over town.
Also the Westin is a much better price point than the BG Ritz and provides the same access to the hill. BG Ritz backs up to chair 16. The Westin backs up to Riverfront Gondola (7) and then Lower BC chair (15) that places you 200 yards above chair 16. Overall, it's far easier to get to BC and the rest of the mountain from Gondola 7 and Chair 15 than going up chair 16 and taking intertwine back into BC proper. Chair 16 also backs up the worst of the lifts on the BG side of the hill. It's usually faster at first chair to take 7 to 15 from the Westin than waiting in the line to go up 16. Chair 16 is a great chair to go up early but there isn't really much on it so it's not a chair you really will lap. It's a beginner chair. 16 is great for access to Arrowhead or back over into McCoy Park otherwise, you're going to find it's a lot more convenient to be closer to Gondola 1. Elkhorn loft is slow but it puts you at the top of Strawberry which then provides easy access to BG via primrose or back into BC via Larkspur causeway into the talons chairs or down Strawberry Park back to Gondola 1.
If you're dead set on staying in BC, I'd look at Borders Lodge which is off the elkhorn lift or If you're looking at Vail, I'd look at Lion Square Lodge. If you have any questions, feel free to reach out.
Edit to add: if you stay at the Ritz or Borders Lodge you will have access to Village Connect to take you to between places in BC but you won't have access to town of avon or vail without a car. Parking at the Ritz last winter was $200/day. Borders Lodge allows 1 car per reservation. The Blue and Res routes in TOA are far less user friendly than the village loop than the in town shuttle in Vail and they aren't free. If you stay in Avon don't assume the bus route will do you any good. If you stay in Vail, the bus route can get you anywhere between WV and EV easily.
If you're flying into EGE, you can take the ECO bus for $4 / person into Avon station or into VTC and then either walk to the Westin or have LSL pick you up at VTC for free.
If you stay at the Westin in Avon, there's a Westin shuttle that will take you into BC or into Vail for free. You can also walk to city market or any of the restaurants in TOA. They also have complimentary suburbans that will take you around Avon and sometimes vail. The Westin shuttle picks up and drops off at Vilar in BC and VTC in Vail. There's also another free shuttle that will take you to into VTC that picks up at Avon Station directly across the railroad tracks from the Westin next to Rocky Mountain Taco.
If you stay at LSL, they have a complimentary shuttle to take you into TOV proper and into West Vail to go shopping at City Market etc. You'll also have free access to the in town shuttle and the TOV bus routes.
I personally think the BG Ritz is the least convenient of the major hotels in the valley. It's super nice but very isolated by intention. If you're looking for that level of accommodations and want something more accessible. I'd say look at the Park Hyatt.
I'm friends with most of the GMs for the major hotels in the valley. If you decide on which one you'd like to stay at lmk and I'll forward you the GMs contact (and I'll let them know you'll be reaching out) and see if we can't get you a better off rack rate than what shows online.
The only on mountain hotel I would personally avoid is the Charter in BC. It's a confusing hotel and it takes some getting used to get home from the hill. It's also SUPER easy to miss the turn off Dorfs Highway between Haymeadow and Gondola 1 and end up skiing along the side of Offerson Rd until you're in BFE. At that point the only way back is to bum a ride from a passing village connect bus. As a tourist, that would make me a little antsy.
Your other option is finding a place at Falcon Point or one of the lodges in Avon off East or West Beaver Creek that's close enough to Gondola 7 to walk. The issue you'll have with that plan is the walk home can be a real *****. The walk to the gondola isn't bad, but the walk home when you're tired and exhausted can be a little bit of a hike. Your best bet if you pick that option is to pay one of the Westin ski valets to let you keep your gear with them. They aren't supposed to do it, but they will if you take care of them.