I am putting together an AR from a Bushmaster complete lower and an upper form Model 1 or J&T.
I want the heavy bull barrel but do I want 16", 20", or 24"? This will be more of a fun gun, but fun to me is never closer than 100m. I would love to have the 16" but am afraid of compromising accuracy.

If you want something for distance yet you want a shorter barrel, do the compromise and get the 20". I actually prefer the 16" and 20" barrel types myself. I look at my guns more as defense firearms though. Of course I plink with them too. If your only use for the gun is long range paper work, just get the 24" as it would be best suited.

Quoted: I am putting together an AR from a Bushmaster complete lower and an upper form Model 1 or J&T. I want the heavy bull barrel but do I want 16", 20", or 24"? This will be more of a fun gun, but fun to me is never closer than 100m. I would love to have the 16" but am afraid of compromising accuracy.

you won't lose accuracy with a shorter barrel.  barrel length in and of itself isn't what contributes to "messier" groups; it's the shorter sight radius with the M4 type handguards (solve this problem with the dissipator model!).

and if it's just for fun, it shouldn't make that much of a difference anyway.

I read on the Quality Parts/Bushmaster's web site that the longer tubes are not more accurate. The longer/short will give you more/less velocity. I think it is about approximately 100 FPS per 1" of barrel lenght. I can't find it on their web site because it has changed. There was once a blurb about what is the diff between a 10, 16, 20, & 24" BBLs

I found it! The link to the Bushmaster website regarding barrels. Check out bullet point 3:
www.bushmaster.com/faqs/barrels-accuracy.asp