If cost is not an issue then these epoxies are an excellent choice. Do the floor first so you can get used to the quirks of working with these products. You pour it on rather than paint it on so you'll have to do a single cage wall, floor, or side at a time. You mix up enough to pour the floor and do that. The next day you could turn the cage on its back and pour that. Make sure the cage is plum and level for each step.
The fact that you have three coats of poly on the cage will mean you won't have to use quite as much epoxy. Many of these products recommend a flood coat, but you can achieve the same thing with poly.
I am a bit concerned with the poly being painted onto the melamine. Poly will not likely stick to melamine and I'm worried some of it will pull up when the epoxy is poured over the top. Again, no big deal since it's the floor which will be covered with substrate.
To be on the safe side, calculate to pour a full 1/8" thick layer of epoxy for the floor.
You likely can get away with thinner coats for the walls and behind the front litter dam, if there is one (hard to tell from the picture).
To achieve a thinner coat means getting the epoxy warm and keeping warm while you work with it. Many people just set the jars in hot water (not boiling) for a while before mixing and then apply it in a warm room. This reduces working time so you'll want to be comfortable with its use. If you're at all concerned just use more of it and don't warm it up.
Sealing the back of the door frames might be kind of difficult.
Also, you only need this for the cage interior, IMO.
Lastly, I believe my friend mists this cage by hand. But I'll ask him.