If its going to be your main hunting rifle, I'd go for a bolt action. In practical terms LH model rifles arent as widely available as RH models and don't come in as many calibre, stock styles or finish options. There are several well known brands you can select from that make or recently made LH models, these include:

Remington M700
Winchester M70 - no longer in production but rumoured to release a lefty in due course
Ruger 77
Savage
Tikka
Sako - releasing a LH model 85 this year

Other than these, there are discontinued brands and some higher cost options, including switch barrel rifles.

If it were me I'd try a few of these and pick the one that handled the best. You can still find LH Win M70's on the used market and pricing is fairly reasonable, unless you want one in 375H&H.

If you have 3k to spend, I probably look to spend 800-1200 for the bare rifle. And then spend the balance on a good scope and set of mounts.