I would try a saddle with a cut out way before a noseless saddle! I get urethral irritation/chafing with saddles without cutouts, including a WTB Speed She saddle with a channel that I was hoping would work, but it made me sting when I peed or was showering after a ride. For me, the Terry Butterfly works well, and no more irritated urethra.

I had the same problem when I moved down here and started riding a beach cruiser with a cushy saddle. My urethra and clitoris were very irritated after riding the saddle for awhile and actually hurt. I bought a cheapie saddle (no name brand) with a cut out (the best I could do here as the selection is extremely limited), and altho my sitbones had to get used to a harder saddle all over again, the pain/irritation "in the middle" of my lady parts went away immediately.

But I do agree with Eden -- kidney pain without urethral irritation or bladder infection sounds strange to me as well and may not be saddle-related.