Signal transmission from FiOS set top box

We finally got FiOS TV in our area and my box of equipment just showed up at the front door. (It's a self install situation since we already have FiOS phone and internet.) I was hoping that, like some other providers, the set top box for the secondary TVs attached to the whole-house DVR would be small enough to fit behind the wall mounted TV in the bedroom. It is, however, quite large. I can mount it remotely (my closet) and use the cabling I put in the walls during construction to get the signal to the TV location, however, I only have the following to work with: two RG-6, two speaker pair, one Cat 5e. Is there a good solution to pump component video over this combination? I was thinking the two RG-6 fitted with RCA plugs could handle two of the three component channels, but can the Cat 5e be configured in such a way to handle the third? Obviously the audio will run on the speaker pairs, but this third line for video is what's stumping me.
Any suggestions?

Ratman, "quite large" is 12.5" x 9" x 2.5". I realize that isn't huge, but it's too large to hide behind the TV. And there is no furniture under the TV to sit the box on. Regarding line levels, I do realize that but wasn't really thinking about it. Obviously, I could switch to using the two RG-6 cables for this audio, but then I'm stuck with an even bigger video problem.

For both Ratman and ctdish, I've looked into doing HDMI/Cat 5 conversion, but it requires two runs of Cat 5 and I only have one.

I'm probably going to try and drill up from the basement inside the wall, which unfortunately is an exterior wall, so I can run whatever cables I need.

Thanks for the suggestions though. I figured it was worth a try.

I just reread Ratman's link. At first I just saw 1 Cat 6 cable was needed. Now I see them mentioning Cat 5e. Hmmmmm. This may be the answer. Thanks.