Lots of good advice and comments here.
Oddly, I've never owned an Epi LP. I did own an Epi G400 SG for a short while. It was certainly gigworthy, though the pickups I found a little bland/biteless. Obviously I could have dropped different pickups in, but I'm not a modder. I sold it as redundant when an Agile P-90 Valkyrie II (their SG-inspired then-model) became available new on closeout for $99. I'd found the Epi G400 in a pawn shop for $175 US, and got that back on resale.
My first budget LP-type guitars were DeArmond M-series MIK models. Both were bought for around $200 US, one new and on used on consignment in a local mom&pop. I had an M-75 (solid agathis with arch maple cap, harp tailpiece) and an M-72 (chambered mahogany with flat maple top and straight peg tailpiece). They were too similar sounding (both having the same DeArmond made-in-USA Gold Tone HB pickups) to justify keeping them both, so I sold the M-75, which was heavier and not as pretty (the M-75 was ebony, the M-72 a burst). Here's a pic of the M-72:
Sadly, I'm selling that guitar as I need the bucks and now have some Gibson LP's, plus I can get a very similar tone from (yet) another DeArmond, my Jet Star (which looks very unlike an LP). But for someone looking for a budget Les Paul-type guitar, I would recommend considering a used DeArmond M-series, which are actually based on the old Guild Aristocrat/Bluesbird, rather than a Les Paul. Build quality is at least equal to non-MIJ Epi's, and you get a US-made PAF-style alnico-magnet pickup you will not have to replace.
I have two Gibson Les Pauls, both of which are humbucker (rather than P-90) models. Both were good deals on the local used market. I got a Gibson Les Paul Studio Doublecut for $525 US (w/hsc), and a Les Paul Classic w/ohsc for $725. These are my 2 most expensive guitars, and I'll probably not spend that much again on any guitar:
Since these are fairly different looking and have different pickups (490/498 "hot" alnico in Studio DC, hotter 496/500 ceramic in LP Classic), I can justify having the both of them. For more vintagey PAF sounds I can use my Jet Star (I don't play the M-72 since I'm selling her and don't want to get too attached), even though it does not look like an LP:
Speaking of sounding like an LP but not looking like one, I'll close with the last of my humbucker LP-type "family" (I'll leave my P-90 guitars for another thread), my Fernandes Ravelle Deluxe. I was very skeptical about paying what I did ($400 US, w/non-Fernandes gigbag) for a used (albeit pristine) MIC guitar, but this thing rocks!
These things come with Seymour Duncan Jazz/JB pickups stock (this is the non-Sustainer model Ravelle). I found mine tagged for $450 US in a local mom&pop and let it sit several months until my birhday came around, then went in, played it and bought it for 4 bills even. I've heard of others getting these used in the $300 US range, but feel mine was well worth the price I paid.