Well, now that everyone else has thrown their ideas up on here, I will add my two cents worth:
* Browning A Bolts are good rifles, but your desire to go to a standard head size cartridge pretty much says it's easier to start over. Not a lot of left-handed actions to choose from, but there are Weatherbys and Remington 700s available and they are relatively plentiful on the market.
* You can buy a left-hand Rem 700 action without breaking the bank simply by buying a used left-handed rifle. Look around at your local gun shops before buying one on line, on line sales are typically higher priced by the time you get the rifle in hand.
* Your "simple" rebarrel job will easily cost you $500-$750, possibly as much as $1000 ($300 for the barrel, $150 thread/chamber/crown, $100 true action, $150 for a trigger if you opt for one, $100 polish/blue if you go that route, +$ fit the barrel to the stock if needed, and add shipping costs). Not a cheap proposition, all in all.
* Unless you opt for the "whole package meal deal", you might just be better off trading in your A-Bolt and buying a rifle in the chambering you want. Although you will not likely find a rifle chambered in 6.5X55 on the shelf, you might get lucky.