why does the hot water cartridge clog up so quickly in the tub?

It's either from the Mineral deposits in the bottom of your water heater or a rusty galvanized pipe.

I would guess the later due to only one faucet having this problem.

learn how to drain your hot water tank . It's easy and gets the minerals out.You need a piece of hose to go from the tank to a wash tub, or floor drain. Garden hose will work. Not the thin really cheap kind. Takes only a few minutes.just open the valve a little,so you won't get burned.

You must get a lot of tiny bits of rock, sand, grit or some other types of minerals in your water. The next time you take it out, see if you can't possibly clean it. My house is 42 years old and I have never even had to replace the origional one yet.

why do you have a filter on HOt water ? it is suppose to be on the cold side