Sounds like you're really into lever actions. You should use whatever you're comfortable with, but I generally don't recommend any out-of-the-box lever action for this application. Unless you have put many hundreds of rounds of your exact "carry" load through it without any hiccups. The actions are more complicated than most people realize, and they can be very finnicky, mechanically speaking. (I have a lever gun specifically for bears, but I haven't worked it over enough yet to put it into use)
If you haven't already, look into the pros and cons of controlled round feed actions. It's not a cure-all, but you should be aware of the differences for your long gun shopping. Many firearms don't cycle well when operated by a panicking user, and you should consider how and if this applies to you. Sounds like you've got a cool head, but it might be different if you or someone you know is actively getting chewed on.
There is also a strong argument in my opinion for a semi auto option in certain cases. If you're considering the Pedersoli, then your budget would easily allow a Browning or Benelli semi auto in a stout chambering, for example. Or certain proven semiauto shotguns.
If you can put the training time in, you may not need a LH specific option, regardless of action type. Can you give us more info about the type of long gun you want to carry? Some people want short/light. Some need a traditional look, etc. If you're in and out of vehicles frequently, then a detachable magazine is a great feature.
For caliber, stick with something common (easy to find ammunition since you don't reload) at 30 caliber and larger. Ammo selection specifics will come after you choose the firearm!