I don't believe I can force a table to be in a particular file when they are all in the same filegroup
That's correct, you do not have the direct granular control over file allocations, only filegroup.
With that in mind, you have a few options. The first one would be to create a new filegroup and isolate whatever tables you need to in order to give you your desired end result of not filling up the drive. Another one, which I'd argue is probably one of the better options, is to put the file in question in a location with more space, or tack on more space to the existing volume. I'm guessing you wouldn't be asking this question if that was a possibility, though.
Another route would be to take advantage of the proportional fill behavior that SQL Server implements (which dictates it'll attempt to fill up all filegroup files at the same time) and size out (larger) the other files in the filegroup. That should translate to a larger percent of the allocations in those other files, as they'll demand more by the ratio of free space (reference this BOL document to further understand proportional fill). Although I personally think that's a hack and a workaround at best, and will only prolong the inevitable of filling up your drive.