Video clocking: crystal or generator?
I'd try to consider the amplitude problem (a 5V oscillation is really pretty high-powered) separate from the clock generation problem.

I think the right way here is, as you kind of drafted, to generate and use a common multiple of both target frequencies, i.e. 12·Fsc, and divide that down as necessary (i.e. by 2 and 3, respectively).

Luckily, that's not very hard:

There's PLLs, that can "trim" a VCO to be a multiple of an input frequency. 

Now, if you spin the idea of dividing the result of that a bit further, you end up with a frequency synthesizer, which might generate a lot of clock frequencies from an input frequency, in "phase-lock". 

TI and many others produce clock synthesizers!

To convert the typically low output amplitudes of signal generators like these, you'd often just use a clock buffer, a voltage translator (74xxLVC series, maybe?), or simply comparators; in fact, since the edges coming out of these devices will typically be very well-defined, and also, because video probably doesn't care about slight assymmetry, but about frequency quality: a pair of transistors would do the trick, too.