Unfortunately there is not a direct way to search by room name. Room names make pretty poor keys in any case - names are not unique, can change dynamically, etc.
I suspect in your situation keeping a cache of the user's rooms doesn't make much sense, so you may want to do some real-time paging/searching - e.g. launch a list rooms request for the user (maybe sorted by lastactivity) in the background and page through a smallish chunk at a time (maybe 20-100), updating your search results UI as the room names come in.