Hello, Welcome, and Congratulations, Beth and James.  

I have a 2351 so the bathroom layout is very different.  Not sure if there is any place in yours that this would work, but am posting it in case it does help either you or someone else.
I purchased a swivel towel holder from Ikea and brought it with me when I picked up my unit.  The factory installed it for me (also the TP holder and a towel hook in the kitchen).  This way I didn't have to worry about screwing something into a bad spot.
I've included pictures of the towel holder.  The two longer bars swivel out from the wall which is nice when I want the towel to be spread out to dry some.  Otherwise, I keep the bars with the towels flat against the wall.  In my bathroom, there is enough room for the towel rack to be mounted to the left of the shower door opening with the bars extending out along the toilet wall.  There is a very short knob below the two longer bars that also swivels.  I hang a tote bag on that (used for supplies when I use a campground shower).
Measurements:  From the wall to the end of the long towel bars: 15 1/2 inches
From the top to the bottom of the mount:  8 1/2 inches
Width of mount:  1 inch
I believe Ikea sells this in two sizes and if memory serves, I have the smaller of the two.
Good luck and have fun!