I doubt that you can solve this in the way that you hope. A LUN's space is always logical. For instance, the backing store for a LUN could go the other way by mirroring itself across SAN nodes, but it will still report the logically available space. So, you have 4TB of logical space with 3.7TB used. The fact that it's compressed and deduped doesn't change logical usage.
What most people do is change the logical size of the LUN. You're monitoring actual use on the back-end, so you can grow its reported space to anything that you feel comfortable with. If you want to minimize the splash damage of overprovisioning that overflows physical storage, you can use multiple LUNs that present more total logical space than physical space.
As to your question regarding full LUNs, Hyper-V will pause all VMs on a full LUN. They are not saved or turned off, but all processing and I/O stops. You can resume them after you free up some space.