My '96 R0 & Set Neck #108:
They are very much in the same ballpark tonally. Of this pair, the LP seems brighter, while the Grosh is more resonent. Not a great fan of Harmony Central, but there's a spot-on review by a pro player of the Set Neck there.
At the going rate for used Set Necks (around the price of a used plain top historic I think) they are a screaming deal, highly recommended. I think a Set Neck would probably satisfy your LP jones.
I've played many LPs, and only one Grosh Set Neck (the one I own), and I'd say the main difference is that that the Grosh can sound clearer and brighter (more articulate), and prettier, but it lacks the deep growl of a great LP. OTOH, I've played LPs that also lack that DEEP growl, but many of them seem to have it. OTOH, I've only played one Grosh Set Neck, so it might be that this one lacks the DEEP growl, but others do not. BTW, despite the lack of DEEP growl, my Grosh is an insanely GREAT sounding guitar. It has a beautiful, open, clarity that I love. And it is a perfectly built instrument. I can't say that about any of my LPs, all of which I also love madly.