Line the crates up on the floor the way you would normally carry them in the vehicle. Then take a tape and measure the overall dimensions of the 3 units. Do this for several arrangements.
You can the take these figures to a dealership and with a tape measure and see if there is a fit. I often do this when buying stuff at Costco, who don’t deliver, to see if the package will fit in my car, or whether I need to ask a find with a truck.
If I had 3 dog cages I would be spending a large amount of time with a sketch pad, tape measure at the various dealers. Sorry about the fried transmission, but I assume that that vehicle could actually acommodate the crates. That’s a good start, comparing your new vehicle with the old one.
Happy shopping!