If you can find a Magic VGA it will take composite and convert it to VGA. The problem is looking at a slightly washed out picture which has the additional problems of signal noise if your PC is nearby. It takes 5V and you can draw this from a handy USB port. It also have PC pass through but no remote. The resolution is maxed at 800x600.

A KWorld1440 converts composite/component into VGA. This device has a remote and runs off 5V, has a built-in speaker and also connects to a PC, has picture in picture which is great for X console to be viewed in a window while on the PC desktop. Resolutions are XGA and do look nice on the projector.

I have both and I recently bought a LCD TV which plugs all my consoles and PC into.
I am currently converting all my soft modded Xbox1s into CoinOPS bartops and these little converters really do help.
I recommend getting this type of device if you cannot find a cheap LCD TV.