Generating two signals with small difference in frequency using one oscillator
I thought about amplifying the signal to generate harmonics, from which I will then filter out the frequencies I need.

Harmonics are always multiples of the base frequency so you'd get 27 MHz, 54 MHz, 81 MHz etc...  That's pretty useless for FSK modulation.

For proper FSK modulation at 27 MHz you should be using something like 10 kHz or 100 kHz distance so 27.00 MHz and 27.01 MHz or 27.10 MHz.

That's not something you can easily do with one 27.00 MHz crystal. Crystals can only be tuned in the order of a few hundred Hertz. So you'd need 2 Crystals like 27.00 MHz and 27.10 MHz and select between the two.

There is also another solution and that is to use a mixer. By mixing the 27 MHz with either 0 Hz (DC) or 100 kHz (or -50 kHz and + 50 kHz using quadrature LO signals), after filtering around 27 MHz you'd get 27.00 MHz and 27.10 MHz (or 26.95 MHz and 27.05 MHz for the quadrature solution).

I would call the mixer solution the "classic" way of doing this.

At 27 MHz you might also consider using a DDS. A DDS is a digital waveform generator. There are single-chip DDS solutions. Some DDS chips should be able to "do" FSK I think.

After a comment about a varicap:

Indeed there is another way. You could make an oscillator which can run at 26 MHz to 28 MHz. Using a varicap we can tune the frequency.

Problem is, how to be sure that it will be running at the desired frequency ? Temperature and many other effects will make it run on all frequencies except 27.00 MHz ;-) . So how do we make it stay at 27.00 MHz ?

The solution is a PLL. Using a PLL we can lock the 27 MHz to a reference frequency, for example the 27 MHz from a crystal or some 10 MHz reference from a different crystal. A PLL locks the ratios of these frequencies, it basically wants to multiply the reference frequency by a certain factor, for example 27 (Fref = 1 MHz, Frf = 27 MHz).

But now the FSK modulation. The PLL has a feedback loop which we can influence. By making it too slow to respond to the FSK modulation the influence of that modulation will be filteered out and ignored. On average the oscillator will run at 27 MHz (the PLL and it's slow loop takes care of this) but momentarily it will be at a slightly lower or higher frequency (due to the FSK modulation). So you can just directly FSK modulate on the oscillator (with the varicap).