Depends. Nothing really sounds exactly like a Les Paul except a Les Paul. I know when I was on my search for 'that' tone I came VERY close to getting a Standard. I waffled a lot and did a TON of A/B testing with my SC245, and realized that while my 245was a tiny bit 'thinner' in tone, it had better overall clarity. That really helped as the gain increased. I also found the tone controls more useful on the 245. I passed on the Standard. Sometimes I think I'll get a Les Paul, but....
Now, how does this relate to your question? I have the same S2 Singlecut you have. It sounds fat and has a lot of sounds available between taps and controls. However, it is not a Les Paul. In my mind, the S2 Singlecut Standard has warmer tone and the big pickguard reduces resonance. For me that's cool. I have a PRS Singlecut without a big piece of plastic on the front. If the S2 is not doing it for you anymore, then go try the LP. Bring the S2 to play next to it. Decide from there....