It sounds like you're trying to come up with a technical solution to a political problem. If you don't have rights to monitor the things you need to monitor, there's not going to be a good enough solution. Just because you monitor file size isn't enough - you also want to know the free space on the drive, and you won't be able to retrieve that without more permissions.
I'd ask the sysadmins if you can give them a stored proc that will be run with sysadmin permissions. They can review the code, make sure you're not doing anything nefarious, and then sign that stored proc with a certificate. They can grant you the rights to run the stored proc (but not alter it) even though you're not in the sysadmin role, and you'll get the results.
Another approach is to give them T-SQL code that they can implement as a scheduled SQL Server Agent job that sends you the results. You get the data, but not the rights.
I'm leery of the sp_msforeachdb approach because if you're not a sysadmin, you're probably not always going to have rights to every database on the system either. If they implement any third party monitoring tools or share the server with more departments, your sp_msforeachdb code may fail.