As you mentioned; the barrel is "your" biggest issue. These Target models are primarily for bench shooting as oppose to shooting squirrels. You can replace the existing barrel with a carbon fiber one. You could also install a stock barrel and free float or semi free float the barrel. It will work but look a little goofy on your existing stock. You can pick up a standard barrel for about $30.00. ...
Or, you can;
Be Safe !!!
Just buy a lighter sporter one with the synthetic stock for carry and basic plinking .