The $600-$800 price range pretty much takes you out of the market for a new autoloader, unless you can find a deep discount on a Rem. 750. These MSRP for about $1000 + tax of course ! Browning's BAR retails for about $200 more.
As far as a lever action, might be able to get a new one in your price range. For sure a good used onme. Marlin makes a nice lever in 45.70 Gov., 444 Marlin and 338 Marlin Express. Any of these should be bad medicine for deer, bear and I suspect moose as well.
If you could stumble into a nuice Brownuing BAR "Short Trac" in 300 WSM or a "Long Trac" in 300 Win. Mag., for sure you are covered !!!
Seems to me that a 308 Win. or 30.06 Spr. using a quality 165 gr. to 180 gr. hunting bullet would be plenty sufficient for black bear. (My experience with bear huntng is zero!) These are relatively common chamberings for BAR's and Woodmasters, as well as many boltactions and the Browning BLR.
If you decide to go with a bolt action, again the $600-$800 pushes the MSRP of most manufacturers to the limit. Especially if you are going to add a decent scope. I do not know of a single maker of modern bolt actions that is not making a good product. I ahve helped sight in dozens and dozens over the past 20 yars and only very, very rarely has one not grouped well.
My suggestion would be Ruger or Savage ..... probably a better chance of staying around your budget. Used bolt actions in good shape are usually relatively easy to come across at reasonable prices. Ruger makes a super "little" bolt action referred to as a compact. A good frined of mine has one in 308 Win. It shoots lights out and is superb for quick handling .... and it is relatively light weight.
And 14 months for a hand gun ? Are you kidding me ?