I've never had any problems with hunting with a heavy barreled rifle. I only have 2 rifles that have sporter barrels,and I can't remember the last time I've shot either one. Just get a good sling,and you won't have any problems lugging it around.
I put a Thumbhole laminate stock on mine,and really like it. Since mine is LH,I had few choices at the time,but today,there are a bunch of choices out there for Savage rifles. Boyd's has a large selection of Laminate stocks in different styles,also Stocky's Stocks.
Mine was for a sporter barrel,so I had to modify the barrel area of the stock to fit.I also sanded the hand-grip area to fit my hand,and fitted the stock to my action.
They have RH stocks for the heavy barrels,so you wouldn't need to do all that.
This is an older pic,the scope/base is different-
There are many different Composite stocks available for them in many styles. Hogue,Choate,Bell & Carlson,HS Precision,Stockade Guns,McMillan,and Manners all make stocks for a Savage. It depends on what you want to spend.
Most are drop in stocks,with very little needed to be done. Some might need to be bedded,but I'd shoot the rifle in whatever you decide on before,and then if it isn't consistent-go ahead and glass bed the action. It's easy to do if you want/need to do it.