Sprayer is good for smaller jobs, so you won't need to mop buckets. But atomizing of butyl stripper makes me cough, plus it tends to evaporate faster.
If they already have a mop bucket at the job, line it with a trash bag and use it. Use their nasty old mop for the stripper too. Throw their mop away when done and give them a brand new one (it needed to be changed anyway, right?). Clean their bucket up really well. BTW - don't use the mop ringer in the stripper bucket, it will just make it really nasty.