Look at other catchers that are good with the bat, lets say Mauer and Martinez before their permanent moves to 1B/DH, they also had about 1/5 games at first and DH.
Victor Martinez Statistics and History | Baseball-Reference.com
Joe Mauer Statistics and History | Baseball-Reference.com
Now tell me, why do managers play those catchers there? Certainly not because they think the bat of the backup catcher is more valuable then the bat of the starting 1b/DH. They might want to keep the catchers bat in the lineup, and that is also why both Mauer and Martinez were converted permanently and I guess Posey will also follow at some point in the future.
However, with the fatigue model we have right now, this move would make no sense. Assuming you want to have your catcher catching as many games as possible, games at first or DH neither help nor remove you from that goal. So if you want to rest your catcher, you rest him, if you think he's more valuable at first/DH, you might as well play him there all the way.
But in real life, I can't imagine at least DHing doesn't rest a catcher.
You're crouching behind the plate for hours four days straight, on the fifth you spend most of your day in the dugout, getting out to swing the bat 5 times, run the bases maybe twice, and after this day he should be as tired as after the previous day? But if he would not swing the bat, if he would be sitting in the dugout all the way, he'd be fully rested?
That is not a good representation of catcher fatigue. I'm not saying that my model is perfect, there might be other ways. But if my catcher is tired from crouching behind the plate and I give him a day off catching duties, I want him more, not less rested.