I think the Handi-rifle without the bull barrel is under $200. Remember, it needs a scope.....
I think Savage makes their model 93 in a combo that includes a scope. Thats a .22 mag. Keep your shots under 100 yards and in the boiler room and it will do the job for you. Ammo is a bit cheaper than a .223 which, when you are 13 is probably a factor. I just looked, and there is one on Gunbroker.com for $240...Item # 15450123.
You can also use your 16 ga with the biggest shot you can find out to maybe 40 yards.
Brian