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Adding Undersized Fuse to Doorbell Circuit?

So recently my doorbell button got stuck and it burnt out both the electromagnetic coil in the chime and the transformer. As I'm replacing everything I was thinking that I could add fuse to prevent a rogue button from causing so much damage again.

The transformer is 16VAC 10W and I measured the voltage as just over 20VAC. The chime consists of an electromagnetic coil that drives a pin hitting a chime (separate coils for the front and back buttons). It claims it's suitable for a 16VAC 10W or 16VAC 15W transformer.

Circuit diagram

Let's say that x amps run through the circuit when the button is depressed. Normally the button is depressed for very short periods of time (let's say <10 seconds for an enthusiastic doorbell ringer). I've read that a fuse can withstand more than its rated current for a period of time depending of how high the current is over the rated current... can I undersize a fuse such that the blow time for an x amp load would be 10–15 seconds? Would sizing the fuse be so finicky as to be not worth it?