Get a half-round, or round, dowel, have it routed on one side the thickness of the cozy coop warmer about and inch or half an inch deep, depending on the dowel size, and stick it on. Or glue it on. I’d have sone concern for a regular foam noodle, but they do make a type that is made to go on hot water pipes, and it’ll be split on one side already. There’s probably a heat-resistant glue that could be used too.
Or my earlier suggestion of filling in the space behind it on that mounting board. It is right against that board, so a little length of something laid right on it would be perfect. So she stands on a wider flat surface. That would be easy. And if you’re not going to change the location you wouldn’t need the dowel / noodle idea.
If it’s a Cozy Coop panel like mine, it’s narrow not sharp. Maybe a centimeter? I’d have to measure. Heck, Queenie when she first came here and was in her quarantine quarters, roosted way, way up on the edge of a fur strip I used as a cross-piece to tape the screening to! I thought she had been taken or disappeared somehow, until I looked up and saw a dark figure overhead. Amazing because it was barely six inches from the ceiling. And those strips are barely a quarter-inch thick. She thought it was perfect. I had covered it with clear tape to avoid splinters (for me) at least. I ended up stuffing a sheet between the fur strip and the ceiling to block it!
Managed to tuck her back in without dropping her. She was last on the handle list, so went to wash hands (before Sherlock could volunteer to "help").
and looks like some lovely vistas in the background 
You still overseas?
