[MC5C]

In a teaching situation, the instructor is going to hope for a full 5 or 6 note chord to ring cleanly. But in the real world, an F chord barre really only needs the low F and the middle strings to sound just fine - no need for the high f on the E string in a strumming situation. Same with Bb - get the low Bb on the fifth string and ditch the high F that gets muted with the barre on the second finger. When you play this way, you soon realize that you aren't really doing a true "barre" with the first finger, you're just stretching it over to play the sixth or fifth string, and your other fingers are playing the other strings. You can add the high B and E string back in with time, it will come naturally.

Edit - you have some really good instruments for learning this sort of thing, but if you really want to learn barre chords fast, get you a nice Telecaster with a 1 5/8" nut and string it with .009's - barre chords just fall out of those things!