Electrical – Doorbell constantly rings even after replacing switch

Wouldn't a slow-blow fuse work? I don't think you'd want to under size it. Just understand what voltage/current ratings you need and find a slow-blow fuse that can handle 10-15 seconds of that.

Something like this. Just figure out where you need to be on the Average Time Current Curves.

Here is my suggestion of what you can do.

Go to good old eBay and purchase a kit that looks like this:

You can find these for very reasonable prices using a search string like "wireless remote control".

The fob is battery operated and activates a relay on the receiver board when the buttons are pressed. The particular model shown comes with a nice looking wall mountable plastic enclosure to cover the circuit board. (Do not mount the plastic box outside however).

Connecting into the existing doorbell is simple. Just access the wire pair between the door bell and the push button and splice in a new pair that will connect to the relay of the receiver box at the NO (normally open) and COM (common) contact points. This connection will allow the activated relay to act as a parallel button with the existing button with no modifications to the existing button.

The relay receiver box requires a 12VDC source to operate. This can be supplied from a surplus AC wall adapter that you could also purchase on eBay or visit your local GoodWill store where you can find a huge assortment of adapters. If you chose to mount the receiver in an attic or crawl space you could find an electrical junction box and mount an outlet cover plate on it to get a convenient place to plug in your 12V adapter.

You could mount the dongle out on the gate directly (the dongles I've used had a sealed inner liner inside the case that keeps moisture out of the inside electronics). What may be preferable is to modify the dongle with a wire pair soldered across one of the button contacts and then run those wires over to a more conventional style of doorbell button that you would mount on the gate. The hacked dongle could then be enclosed in a plastic hobby box that is sealed and then mounted to the exterior in an unobtrusive area while at the same time the plastic box letting the wireless RF signal come through the box and make its way to the relay receiver board. You will occasionally have to replace the dongle battery but they seem to last a year or more with moderate use.

I've used almost identical kits to replace my very old non-functional garage door openers with the relay module wired across the manual open/close buttons that are inside the garage. The dongle is then easily carried around or kept in the automobile.