A small hand snake usually will not go through too many bends before it just starts to spiral around in the pipe. I would be surprised if it actually made it out 15 ft. A stiffer cable is usually necessary to get any distance.
While in most states I've worked in, 2" was the minimum size by code, here in WI, until recently, 1 1/2" pipe, with just a standpipe, was legal. Inch and a half pipe, when fully open, and installed correctly will keep up with a washer, but as it builds up with stuff, it'll back up.
If there is room, I would go the route of a laundry tub. If (or better said: when) the drain starts to slow, you will have capacity in reserve. Also, I can't speak for everywhere, but most states wont allow an air admittance valve on a washing machine drain (with a few exceptions), because positive air pressure can be an issue.
The tougher part is only having a 1/4 in of pipe stubbed out of the floor. Make sure there isn't a trap in the floor as well.
sorry for the rambling/incoherency of my reply, it's been a L _______ O _________ N _________ G day