I also travel with my dog so I completely understand your dilemma! Here's what works for me - "baby" in the carrier on the floor..I put my handbag (35) in an H golf bag. It's sturdier than a Longchamp or a SportSac, although it is still soft sided. The handles do stick out of the top, but I cover them up so that they are nice and protected. Also, it has a nice wide strap so you can carry it over your shoulder. Voila! Little Misto and Birkin can travel together!
I have packed my Birkins and Kelly (not all on the same trip!) in my roller bag carryon. I stuffed it full of underwear, socks, jewelry, strapped it up, and padded it all around and through the handles with my clothes. Nothing gets squished, and you're not really losing any room inside the suitcase. Since I'm the only one handling it, I know everything will be in exactly the same position as when I packed it. Except when you travel internationally. The customs guy who opened my suitcase on the way home from Paris was impressed with how well packed my Kelly was inside the suitcase. Sadly, he did not do as good a job at repacking it...
Edit to add, I would never put my H bags in the overhead compartment in a soft sided bag. I've seen how the flight attendants and other passengers squash things up there.
Thank you both so much for your ideas!
salazac, that H golf bag sounds wonderful! It didn't get mashed up by other people's suitcases smushed in there? I have never seen a H golf bag, I will definitely check it out

ouija board, that is the exact reason I don't feel comfortable packing my Birkin in anything that isn't a hard luggage piece-- other passengers are brutal! I have cringed so many times with chanel totes i packed in soft-sided LV luggage, so there is absolutely no chance I would do that with a B. When you open your suitcase, does yours have those two almost, ribs in the bottom? It seems like my Pegase 55 rolling bag stores the "handle tracks" within the bag-- so the inside lining at the bottom isn't flat. (that may be confusing to visualize) Do you have a better suitcase recommendation? Of course I want something that could fit the B and ideally a couple other things, maybe jewelry or a pair of CL's I don't feel comfy checking.