IF ... IF ... your institution has a private network that the 4 hospitals all have access to, what I might do is to feed each weather station into a separate installation of CumulusMX running on a Raspberry Pi, and simply publish those pages to your internal network, with no additional security beyond what your network admins have already set up.
Each CumulusMX would be a separate webpage, but just name them similar names, lets say "Weather Mike", Weather Jill" Weather Bill", "Weather Sue" etc except use names of the hospitals.
A drawback is that the CumulusMX Settings / Admin page is not secured, so even though the viewers were all internal to the hospital network, changes could still made. If this is a concern, the data could be published to a separate internal webserver with some more effort.