If you assign the program types and construction sets to the rooms in the last step of exporting the rooms, they should be exported to gbXML and OSM. I can see that you have already figured this out, but here is a screenshot for feature reference.
See this post for creating custom constructions. We currently do not support reading the constructions from Revit directly, but you can make similar constructions as a construction set and assign them to your rooms.
Hi @kisuouou, welcome to the forum!