Bottle brush is better than nothing. Construction supply places have brushes for cleaning out holes drilled in concrete prior to epoxying fasteners in. Some are nylon bristle but there should be metal bristle ones too.
Problem is, the BAFFLE hangs from the top of the flue. The best way to clean the flue is remove the Burner at the bottom, then go to the Roof. Pull the Baffle and clean with Wire "bottle" brush down from the TOP. Clean up where the Burner was, make sure no flakes in anything, reassemble. Don't forget the Baffle. But at least on our coach, even when I take the rain shield off of the fridge vent, the screen is one piece with the rest. I'm unwilling to remove the whole vent assembly from the roof.
So... I do what a tech suggested. Reach up in from the bottom, past the burner to where the flue extends above the insulation pack. Rap on the Flue with a Screwdriver Handle. Pull the Burner and clean p whatever fell down and call it good.
Thinking about this, if the top of the Flue clearly lines up with part of the plastic "mesh" of the plastic Vent Cover... Cut a section of the mesh out, reach down with coat hanger to fish the baffle up. Clean Flue with Brush extended with Coat Hanger. Lower Baffle back in and be sure the wire hooks to the flue top. Reattach the cut mesh. ZIP Ties, wire, little metal bracket etc.