To summarize: Since the Si5351a's load changes at TX startup, its own power dissipation changes substantially as others have noted. In other words, it warms up. So, when the internal oscillator's load capacitance changes in response to the increased temperature, it pulls the crystal's resonance frequency. When using an external oscillator instead of the original crystal, that changing capacitance is isolated from the oscillating circuit, eliminating the effect.
I would like to offer an alternative theory that I think also fits the data: The changing temperature of the Si5351a at TX onset puts some heat into the copper PCB traces that go over to the crystal. The crystal then warms up a little bit and generates the observed chirp. The PCB itself also moves some of this heat to the crystal, causing it to drift frequency a little bit during the entire TX. Additionally, heat from the PA transistor could find its way over to the crystal, also contributing to drift during the TX.
Regardless of whether the internal capacitances are changing in response to the temperature rise, or the heat actually reaches the crystal, the solution is to use Park Mode to reduce the Si5351a's temperature changes. I added a capacitor load to Clk1 on my U3S to mimic the load of the PA in an attempt to make Park Mode's dissipation more closely mimic that of TX mode.
If the latter theory is contributing part of the drift or chirp, thermally isolating the crystal could reduce the effect. Once can lift the crystal off the PCB adding very thin wires to connect it. Those wires would reduce heat transmission from the Si5351a. One could glue a chip of metal to the crystal to slow down the heat gain from the PA. One could wrap the crystal with foam to reduce the heat it receives from the PA.
If the former theory is dominant, maybe the effect could be reduced by programmatically minimizing the Si5351a's internal load capacitance and adding external NP0 capacitors on the PCB to give the crystal the load it expects.
Any thoughts on this?