How to switch composite video signal inputs to one output
Yes, it seems feasible.

The bipolar transistors will not be a good solution. There are other ICs that could do this job, but this chip NJM2595 seems to be a very good and proper solution. About 10 years ago I tought about doing a project like this, but at that time I didnt find a IC like this (I tried a chip DG411 that didnt worked well with video, so I "closed" this project). 

I would suggest that you download and read all the datasheets of other chips of the same function, even if you already decided for NJM2595, because those datasheets can be instructive and give good design ideas.

Of course, you'll need a custom built PCB (don't use perboards; video signal is critical), but as this chip needs just a few extra components, this will not be a very complicated PCB - but this is about your level of experience. 

EDIT: Since this is a SMD IC, consider about using an adapter PCB (search for "breakout board") like this, for an easier soldering of the final PCB:

Google about the PCB layout requirements, e.g. take a read of this whitepaper (AN-6041 PCB Layout Considerations for Video Filter / Drivers