$2K AUS ($1500 US) will get you a great guitar. If you're looking at new, the higher end options like a Gibson Les Paul Standard/Traditional are a bit out of your range, and the same for the classic PRS McCarty types. But, you can get into to Les Paul Studios, which are minus a handful of frills but pretty nice, or something like a PRS Mira. All very nice.
Or you could work the used market. My first guitar was a used Gibson Les Paul (in 1982...and what an idiot I was to have gotten rid of it back in the late 80's!). Important things is, you do get more guitar in the used market than new. Maybe not as shiney or may have a few roadworn blemishes...mine had many and a huge buckle rash on the back. And consider people will now pay
for a new guitar that looks used, just get used, ya know. Used will get you more.
It's hard to say what brand/style will work for you. It looks like you're looking at a humbucker style so many options there. And really, it doesn't matter the brand. If you can, go play them. I've play Les Paul's that I've hated. I cannot enjoy playing a Les Paul Custom. Just can't. Get me a Les Paul with a 60's style slim taper neck and I can play that baby all day long.
That's kind of the key too. Know what 'feels good' to you. I know I don't have the longest of fingers. Average-ish, so I don't much enjoy thick necks (though that's a lie to, my ES333 has a 50's round neck and I love playing it so there's that). Go play them and have an idea what you like so the salesperson can help guide you. Know what kind of pick-ups you want. Do you want hotter pick-ups or do you want more mellow pick-ups like Gibson 57's.
Know what is comfy for you and the ultimate goal of your sound and that will do much to guide you to your near ideal guitar.