For a given barrel profile, such as 15/16 X 32 as on a GPR, the smaller the caliber, the heavier the barrel.
Not only that, but incredibly heavier.  Dixie Gun Works catalog has a chart in the back which spells out the weight of octagonal barrels of varying sizes with bores of varying diameters. 
If you really want a smaller caliber rifle, drop ins are easily made up.  I have a .40 in my Seneca which requires a 13/16" barrel.  It was .45, but I had that barrel reamed to .50 smooth.   I also have a .45 factory barrel to go into my T-C Hawken which requires a 15/16" barrel.  Years ago I made up a .30 barrel to go into the same gun.  To get the  15/16" .30 barrel to balance, I had to cut it to 24".
Before anyone asks, yes it is a muzzleloading .30 cal. barrel 15/16" across the flats.  Rifled 1 turn in 48", especially for round balls.   I shoot either .285" cast balls, (Boy ws it fun finding that mold!!) or .27" # 4 buckshot.  Either works as well as the other.   12 gr. 3F Goex for both.  With homemade Tap-O-Cap caps it's nearly free shooting and keeps a kid occupied for hours.
Three Hawks
I strongly advise you to get a small caliber rifle with a slim profile barrel.  You will be much happier for many more years.