Gail beat me to the "change to Full recovery model and do Log backups instead" option.

As far as your proposed "full after each snapshot" option, you can try looking at extended events, as well as seeing if Windows Task Scheduler can trigger off of the snapshot (then you use sqlcmd to run msdb.dbo.sp_start_job), but the brute-force way is to set up a job that polls every N minutes in about the right timerange, and compare the msdb list of snapshot backups to a stateful list of prior snapshot backups. When it sees a new one, kick off a full backup and record the new snapshot.

Personally, I'd have to say that Gail's options are the best - your Full backup after the snapshot may fail for a variety of reasons, leaving your every 4 hour Differential backups silently worthless.