I've done that a couple times, usually it's an extra hundred a week, which would be fine. This is with "Payless", which in my experience is only a so so rental car agency. I prefer Enterprise for various reasons, but prefer saving money even more.
The Hyundai has a sunroof, much as did the GTI, and neither get opened often at all. I suspect if we had a convertible, the same would be true, even more so.
When my first CTS got long in the tooth, the sunroof would at times stay open, and it happened when my now wife was driving it and it started to rain, she panicked, I had to coach her onto how to close it. She did not like that car much, it had a tendency for the back end to come around when accelerating (which she liked) and turning, especially in the rain. Over steer. Nice to have on a track.