What airline are you flying? There is a lounge in both international and domestic. Both are Priority Pass lounges. The international lounge gets very crowded late in the evening when many of the long haul departures are and Priority Pass customers are often told to wait until a certain number of people leave. Usually though if you have an invite from an airline you're still allowed in.
I have only flown UA and CM out of LIM. CM I was in business class and I'm pretty certain did not give me a lounge invite, I had to use Priority Pass. UA always gives me a lounge invite regardless of which class I'm in and I would assume they would do the same for other airline *G holders. I can't speak for AV's policy if that's who you're flying, but as you noted it's not a Star affiliated lounge, so it's up to them if they give an invite or not. I'd imagine that you'd get one in business class for sure, but I'm pretty certain CM did not, so who knows. I have no idea why AV doesn't operate their own lounge being that it's a hub for AV Peru, but to that point I don't think LA have their own lounge either. Other AV lounges I've been to in the past are pretty basic, so maybe for the better in the end. (never been to BOG, just some outstations).
Also worth note, this lounge is an oddball. You only get 2 free drinks from the bar, they stamp your boarding pass to verify this and it's also stamped on entrance to show that's the boarding pass you entered on. I tested it last time I was there to see how closely they enforce that and used a duplicate boarding pass I printed on the kiosk and was served a drink with that one, but they looked at it really strangely because it didn't have the entrance stamp on it. I don't expect that would work consistently.