While some type of on/off switch is the easiest way to cut off the action, here's one way to implement a time delay.
For less than $5 on eBay you could pair a 1) voltage converter from AC to 12V DC, and 2) a Delay-OFF relay module. The Delay-OFF module will apply the AC voltage to the accessory for some settable duration (like 3 minutes) and then cuts power.
Separately, I'm not clear on why the AC 60 Hz buzzing stops after the coil is continuously powered for 30 seconds or whatever, but if you want to drive the coil with DC some variation of the AC-to-DC voltage converter module could be used.
And if the mechanism maintains its position with a reduce coil voltage once it goes down, note that a time-delay relay module could be used to switch in full voltage for the first 5 seconds (or whatever) when initially triggered...and then switch in a reduced voltage thereafter.
There are all manners of low cost electronic modules on eBay that are just a few dollars each where you can cobble together some interesting functionality Lego-like without soldering.
Another approach is a thermal cutoff switch. These are about $1 and you choose the cutoff temperature like 70C, 80C, 90C, whatever. Place next to the coil and wire it in-line with the coil. When switch trips, it cuts power to coil until the temperature drops and the switch resets; not to be confused with a thermal fuse which trips and game-over until you replace it.
