Covering roof vents to keep out roof rats

I cannot say I'm a fan of the design of that vent - although looking at your other question, I'm rather surprised that lint is falling down into that gap. Possibly your fan is creating too much pressure, and it's backing up inside the vent at the 180 and thus air is escaping through the pushing some lint down into that gap, where it can get stuck.

If that's the case, then you might try installing a baffle inside the vent to force lint up far enough in the vent so that it has to go out the intended port, but which still allows water out. Like so...

Sorry, the color doesn't show well, but inside I've added a piece of metal running from just below the intake port to nearly the end of the angled portion of the vent, with a brace connecting it to the body to keep it stable. You can't see it but there should be drip holes at the point where the new plate hits the flashing for water drainage.

The idea being, lint flies in and cannot be forced directly down to the gap in question, but water will hit the tray, run back and go through the drip holes.

That or replace it with this

My experience using those brushes is that it's difficult to tell if you're pushing on a lint blockage, a turn in the pipe, or your knocking the cap off the end of the pipe. So if at all possible, you should have the end open and/or connected to a vacuum. You don't want this tool ramming into the motor in your dryer, nor do you want it knocking the damper off the outside of your house. In your case, that's going to require someone on the roof.

You can get close, keeping track of how many segments of the cleaning tool you used to get to the roof in a previous cleaning, and then stop a few feet early. But as Michael says, the cap itself should be cleaned since lots of lint will build up there.