Agreed, the Mercure on Welshback is a bit of a challenge to get past reception, unless they're really busy.
Stayed there once or twice when work's sent me late to Bristol on evening jobs, it's a tricky one.
The Thistle hotel is not far from the Mercure and easier to slip in, as is the Marriott, but both are a bit pricey.
Ibis (Harbourside or Temple Meads) is better, just be aware the Temple Meads one has a bouncer on the door on Friday nights and probably weekends too, to keep out the p*ssheads. You need to have either a doorcard already, or look like you're about to book in.
Don't even bother with Premier Inn on Queen Square, reception's impossible to avoid.