Not all SATA controllers support hot-swapping. This is not simply a matter of having a hot-swapping enclosure & switching to AHCI mode. More than likely, the motherboard is beeping because (despite usbsafelyremove's claim) it did not expect a hardware failure. (yes, losing power to the motherboard looks like a hardware failure) Even if Windows is OK with having the device removed from the system, the controller may not be happy.
All that aside, you might be better off looking for specific drivers pertaining to your motherboard/SATA controller which may offer a better interface for removing drives while the system is running. Frequently, there's a simple RAID utility that can perform that task for you without having to turn to some random 3rd party app. Additionally, it is probably a good idea to look to the manufacturer of the motherboard to see if your motherboard does support hot-swapping.