According to the datasheet that Andy linked to, the chip can be configured for 0 dBm output. With the proper antenna, that is probably good enough for 20 m, but you'll have to test it yourself to know for sure. If your environment is noisy, or there are obstructions causing path loss or multi-path interference, then it's hard to say how much power is required. The relative alignment of the antennas also matter. This is due to both radiation pattern and polarization.
Another issue with a separate power amplifier is regulatory compliance. You need to look up what the maximum allowed transmitter power is in the band you are using and for your purpose. If that's 0 dBm, then any additional amplification would be illegal, for example.
Is there a way to calculate the approximate range of this remote controller?
In theory, for infinitely open space, yes. This would assume particular antennas at particular orientations. However, that's generally not very useful.
The real environment is too complicated to quantify easily, and usually unpredictable and changing anyway. There is no substitute for testing in the field.
That said, my gut feel is that 0 dBm at your frequency with properly matched and oriented antennas that have dipole-like radiation patterns really should work over 20 meters. But, if there's a wall or other stuff between the antennas, or one device is hand-held so that antenna orientation is arbitrary, or there's interference in your band, then you just don't know until you try it.
I would start with the 0 dBm output, then only add an amplifier if tests show you need it. Again, make sure you know the legal power limit for your band and purpose, and be sure to stay within it.