SGs tend to have quite slim necks though, not really chunky. Most of the Epis LP will also have more generic slimmer necks (compared to a '58/'59 LP style one). You may struggle with getting a thick neck at that price point, as most lower-price guitars do have necks that aren't particularly thick with a fairly standard C-section rather than a D-section. You may find an older PRS SE , but those neck tend to feel chunky because they are almost all 'wide and thick', and you don't want the 'wide'.

Some of the basic ESP LTDs are very well made guitars (and should certainly be found in that price range used), but are (IMO) desperately in need of a pickup replacement - which is ideal for you. But the necks are still that generic C shape.