You will want to remove the switch as it will be in the way to remove the wire and replace. It's possible to do without solder, but you would need great finesse and patience to guide the wire in and out of the slot. The stabilizer itself just clips on the plate. Leopold uses a special size on their spacebar, and I don't think it's available anywhere. However you don't need to be concerned, as I assume just your clips is broken so you can reuse that wire from the original one.
Part you need:G99-0224 or G99-0225, whichever is easier to find 'with frame' = plate mount. It's the wrong wire size, but like I said, you can pull off the original one and just use that new part to replace the broken clip.