Does anyone know where I can get a guitar like that without having one custom-made?

Why not just a regular Les Paul-shaped guitar equipped with either a FR or TOM?
I've never liked the TOM. bridge and I lock the tremolo whenever I use a guitar with a floating bridge.

If avoidable: Active pickups
I've seen some Les Pauls with a Hipshot BabyGrand Bridge, but I'm not sure about the brand/models and I don't know if they had TOMs before and the owner swapped it out for a Hipshot bridge...

Some people around here may be able to give you some info about that, but if all else fails, you could look into PRS guitars. They look enough like a Les Paul and they have their own fixed bridge. Just an example: http://www.thomann.de/pt/prs_se_standard_245_tb.htm
Last edited by DanyFS at Nov 2, 2015,