When I was first learning to climb double-rope, I experienced the same problem, and I was puzzled about what I was doing wrong when I was sure I was following all the instructions and requirements of this industry standard setup correctly. The fact is, you're not doing anything wrong. The hitch interference you are experiencing is a common problem with that setup. You have several options. You can do leave the setup just as it is, be mindful of it and just get used to it. You can connect the end of the rope to your bridge instead thereby losing the self-tending feature, but you can restore the tending by putting a small Prusik on the rope opposite the pulley and connecting it to the pulley with an accessory carabiner. Or you can run a strap over the shoulder to tend the pulley. There are other options, but, generally, I would recommend that you get familiar and comfortable with whatever double-rope/MRS setup you like and then get a Rope Wrench and start climbing single-rope/SRS. You will be a much happier climber.
Also, I am concerned about your "permanent TIP." If you are leaving your rope setup at all times, be aware that squirrels and rats could chew on your rope where you don't see it. Instead, just install your rope each time and benefit from the practice.