The power amplifier, when keyed, causes the main HT supply volts to
change at a rate which depends on the size of the HT smoothing
capacitors. This change in HT volts also affects the DC supply to the
oscillator tube which causes the frequency to change in sympathy with
volts. Most tube oscillators change frequency with supply volts.

So stabilise the DC supply voltages to the xtl oscillator -
especially the screen-grid DC volts. It won't matter if the DC supply
volts are increased by 15 or 20 percent in the process.

Use a gas-filled voltage stabiliser tube if you can find and fit one.
They used to be available in voltages from 50, 75, 90 to 150 volts.

But a few transistors could do the job.

Adding caps to the power supply will merely slow down the chirp and
perhaps make it more noticeable.

If the DC supply to the oscillator is already stabilised then there
may be a small amount of RF feedback from buffer or power amplifier to
the oscillator. In which case you have a much more complicated
problem to solve.
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Reg, G4FGQ