Counterbalanced just means the dog's rear end weight sits behind the wheel, not in front of it or on it, and thus helps by lifting the weight off the front end. You don't need front wheels, just a front support band which allows the weight of the tush to lever up the front a little. It takes a lot of weight (Eddie's says 1/3) off the front.
It's correct the Eddie's front wheels have to be put on after the dog is in the cart, and they don't swivel. Not an option for your dog. If you were close enough to someone who had the right size counterbalanced Eddie's cart to try it, though, I would.
Doggon's full quad is not all that different from their front-extension. (Eddie's is, it is meant for quadriplegic dogs.) Doggon has a couple of things- one, the soft saddle tends to be nice for dogs still using their legs a lot. Your dog might be okay using back legs if partially supported. Adding the front extension later does cost just as much as getting it in the first place or getting a quad cart, but it allows you to return the cart for a significant refund if it doesn't fit. The downside on Doggon is that they have less of a counterbalance so it is more likely your dog WOULD need front wheels. What the counterbalance would do for a three-legged dog is move the weight behind the wheel so your dog could still hop in front. I have no idea if that would work well, though, and wouldn't want to buy a cart without some kind of guarantee.
So the short answer is, I don't know what I'd do. Eddie's with a full counterbalance MIGHT work, though, and you might find one of those used. You could always resell it if it didn't work.
I should mention that if you go full quad, I'd also look at K9 carts (either company.) Their front wheels are nice- not all-terrain, but closer to it than Eddie's and easy to use. For example this front-extension cart looks pretty good.http://www.k9-cart.com/Pet-Wheelchairs/ ... e.tpl.html