EMG RPC does it nearly-perfectly, but to stop it getting too 'Stratty' you'd need to only turn the control to something like 15-20%, and at that point you're converting your guitar over to an active system for a control you're hardly using... seems a bit pointless.
A less effective but cheaper and easier method would be to disconnect the tone controls from the guitar, and replace the volume controls with 1meg pots. This will bring back a lot of brightness and 'snap' to the sound, not entirely dissimilar to the difference between having a rosewood fretboard and a maple one.
You should also try lowering your pickups further from the strings, and consider using lighter strings. That'll help thin out the tone a little.
Lastly, you could replace the bridge saddles (or even the entire bridge) with titanium or steel versions, and make sure the bridge tailpiece is locked down as low as possible. Those will also add more snap and brightness to the sound, more like a fat Strat.
Of course, it does also depend on the Les Paul in question, as well as what kind of fat Strat you're thinking of. It's very possible to get fat Strats which sound fuller and warmer than Les Pauls, and vice-versa.