Use the Cat5 as a draw wire,& pull a
real video cable through!
Seriously,though,it is quite commonly done.
A balun would be the best idea,as you need to get the benefit of the balanced pair.
You
can lose the DC component in a video signal & restore it at the other end,or get away without restoration.
You may,in fact, find that the Picture Monitor you use has a DC restorer built into the input circuit,so try AC coupling first.
If you do need it,DC restoration can be simple or complex----you probably can get away with simple.
Google for DC restorers & Video clamp circuits.