I have had coon hold up in a house for 2 weeks before going back out through the roof vent when it's cold and snowy
Very common. Coons will become inactive during really cold spells, sometimes for 2 to 3 weeks. If, as mentioned in the original post, this is truly a long-term coon problem, the need for an immediate solution/removal may not be critical.
I would get that one-way removed, and take it to the local recycling center (scrap yard) where it belongs. Mount a 2-door trap (positive set) over that opening. If it takes a week or so for the coon to move, so be it. You'll get it when it ventures out.
If they are suddenly demanding it needs to be out ASAP, if you can see it while in the attic, a quality pellet gun should do the trick. Probably not legal to discharge an actual gun if it's in town, but pellet guns (such as a break-barrel) should be legal.
And if the owners don't want it killed.... they obviously aren't serious about wanting it out. If they ARE serious about it, yet refuse to use lethal methods, they should not resist the cost of a lift (if needed due to snow on roof) to place the posi-set at the hole.
You are basically fighting a couple obstacles with such a job. One: Mother Nature which is in total control on the raccoon behavior. Two: A client that is demanding results but possibly putting limits on your efforts. You can't control #1, but #2 is something that can change.