I have one exactly comfigured as your's. BC Stock and BC lightweight barrel. Built it specifically for my niece (10 yo) and wanted it as light as possible. With scope it is ~5 lbm. An OK gun but the barrel is not very accurate. Best I can do with match grade ammo is 1" (10 rounds) at 50 yds. out to 1½"-2" with bulk ammo. Even inletted the barrel channel to allow free floating (not a big issue for the alloy receiver due to the light weight of the barrel). Groups no better. Great for a general plinker but not so good for accurate work. I seriously doubt my niece would notice the lack of accuracy. Then again, it could just be just a lemon barrel. Your's might be much better. Others at rimfirecentral have seen similar result as me, though.
If you want something a little more accurate but not for a whole lot of money, get a Green Mountain all steel barrel. Got one on another 10/22 (Bell & Carlson stock) and it shoots 0.5"-0.75" / 50 yds including bulk no problem. Sometimes they run sales for them at rimfirecentral.