Unlink the remote take it to another location and link it with a Joey if you have a joey and do the same with the Joey remote link that one to the hopper and see if it does the same. Make sure to unlink it from the other receiver. If it messes up while linked to a joey probably the remote.
Could be bad lnbf, Signal drifting in and out. If you have your dish on a pole and has black cable in the ground could be the problem it needs to be replaced to orange burial cable. Bad cable, bad connector/ground block.
Put a splitter and run two joeys you can stack spliters its fine. I wouldn't put a tap where you described it'll be a headache. The super Joey requires its own line from the dish due to the frequencies that the super Joey needs, it adds two tuners to the system. And you'll have to tie the...