Quoted: Is there a way to install a bull barrel without a free float system?

Not without a custom built handguard cap

But I got to ask: why?  What do you think you're going to get from a bull barrel that you can't get from an HBAR?

. How does the hand guard cap fix itself to the front sight?

It doesn't.  It just sits on the barrel and the gas tube prevents it's rotation. Most barrels have a ledge that is thicker than the .750 of the gas block/Front sight area to prevent the handguard cap from sliding toward the barrel nut (and the front sight prevents it from sliding foward).

Am I going about this all wrong? Will I get all of my parts and find that the guards won't fit a bull barrel?

I have a few suggestions you might want to look into.

1) Use an HBAR instead of a bull barrel if you really want a heavy barrel.  You won't find any difference in accuracy (especially from a non free floated barrel).  This will allow you to use the handguards you want.

2) Instead of trying to use Airsoft non FF handguards buy a YHM Free Float 4 railed handguards.  For not much more than the Knock-offs you can have a free float system that will work with your barrel and help with accuracy (about $90 for the unit).

3) Contact Kurt Wala of Kurt's Kustom or Randall of AR15barrels.com and see if they can modify a standard handguard cap to work with your bull barrel