This works by turning a pin on and off very quickly to generate a carrier wave. However, this produces, not a sinusoid, but a square wave. This means if you attempt to transmit on 1khz, you will also transmit on 3khz, 5khz, 7khz etc. So you will certainly end up transmitting outside of the band you were intending to.
Of course the true advice should be "don't transmit on an antenna at all without the appropriate legal license".
Antennae tend to give a multiplication factor to almost any input signal, no matter how low the initial power level might be. As a crude example, a 1 watt input signal might suddenly have an Effective Radiated Power in the hundreds of watts (yes, there are a vast number of variables in such a formula so take this as just a thought experiment).
A bandpass filter has the effect of only allowing a desired sliver of the RF spectrum to be transmitted beyond it (i.e. Block All Except For...). There are cheap, passive, dumb analogue bandpass filters commonly available, and at the higher end of the cost scale there are active, smart digital circuits. Do your homework because not all bandpass filters can provide the sharp shoulder needed to give a clean slice of spectrum. A similar device is a bandstop filter, which blocks a specific slice of the RF spectrum as specified (i.e. Allow All Except For...).
That multiplication factor is easily negative though, unless you specifically design the antenna to have gain. And a 20dB gain like you are referring to is a very high gain antenna already that will have to be designed and built quite carefully.
My point was that it is ridiculously easy to get it all wrong and suddenly achieve unintendedly strong radiated power. Again, it was a crude example of a thought experiment with a wide, wild number of variables to be considered.
I'm guessing this generates a really dirty signal, lot of harmonics, etc…
So not using a bandpass implies that you are polluting the spectrum which the FCC would obviously not approve of but that can't be it because none of this would be something the FCC would approve.
The FCC and similar authorities worldwide provide slices of RF spectrum for amateur experimenters. Even so, there are rules that require your homework and attention.
Could you do amplitude modulation at RF frequencies with a resistor ladder DAC using the pi?