Why not get a round replacement one? Or get some work lights if you want more light and wire to come on with reverse ?
 
I've had LED reverse lights for years (along with the rest of the lights), no problems and super bright. Personally I prefer the look of the NAS type (Le Perei) rather than an aftermarket, still not standard but I like them.

 
I have an LED work lamp and the clean white light is very good and I use that for reversing but have to switch it on...it would be a fair bit brighter than what you have, the white light does that.
 
You may well need to install resistors as the draw on LEDs is far less than regulae bulbs.

Mine started turning every light in the car on!
 

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