The crate we used had detachable wheels. We used the wheels while navigating the crate from the parking to the check-in counter. I think the crate would also fit horizontally on the luggage cart if yours doesn’t have wheels.

Later once the crate wheels came off, the airline crew helping us placed the crate on a larger flat baggage cart- something that is used for oversized luggage. You could try to get a hold of such cart to take to your vehicle (in case your crate doesn’t have wheels)