I carry 4 bikes on a receiver-mounted bike rack, an 'L' shaped one as well. Two are children's bikes, and 1 is a woman's bike (DWs bike).
Before I got the bike rack, I read all kinds of horror stories about all the trouble I would have fitting these bikes on the rack. It was a challenge until I got the right rack and figured out how to do it, but they all fit on the rack fine. You just have to get a little 'creative' sometimes in where, how you strap them on. :)
For DW's bike, and both of the kid's bikes: Kinda hard to explain, but what I did was, in the front of the bike, the very bottom bar of the frame sits on top of the rack/saddle, and strap goes around it. For the back, the saddle, bar goes in the opening between the vertical bar going down from the seat, and the bar going back to the back wheel. I turn the saddle sidways, and strap to the vertical bar. Works great, although I will confess I bought and returned 2 different bike racks that wouldn't work, before I got the one I have now that works fine with this, a
Thule Trailway 4 bike model.
Anyway, not sure if you might be able to mount them like I did, but just thought I'd throw that out. I definitely understand your frustration, I just went through it. It can be a pain finding a way to mount and carry multiple bikes with different frame types. Good luck with it. :)