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Also, I noticed that the effect is increased if the mount screws were only loosely tightened.
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I wonder if some of the effect could also be due to mechanical (50Hz) vibration in the transformer that causes microphony in the VCO? Sometimes it only take a tiny amount of 50Hz vibration from a transformer to make its way into a VCO. At work I used to test all my VCO designs on a vibrating table. It could be driven with various waveforms to replicate vibration in vehicles or from shocks etc.
I also had another look at your earlier VCO tuning data and 11.9V Vtune at xx.999MHz seems very close to the 12V supply limit. Maybe the VCO slug needs trimming slightly to give some margin here. When it gets this close to the supply voltage it can affect the loop response. It should tell you in the service manual what voltage is needed here and I think it would be best to adjust it. Obviously, take care not to crack the tuning slug as it could be very brittle. I would use a plastic trim tool here and not a hard tool.
Obviously, you have to make sure it doesn't get too close to the 3V Vtune clamp set by the zener diode when it is at the other end of the range. As I said before, I think the zener diode by the TC4016 PD switch IC is there for startup reliability. If the Vtune voltage could swing very low at startup I think the VCO frequency could get low enough to produce a (38MHz-VCO) mixer term that would be too high to get through the post mixer LPF in the PLL. This is the LPF shown in blue in the amended block diagram below. If this happens, the loop would be broken with no feedback signal to correct itself. So I think the clamp diode is there to provide a hard limit for how low the VCO can tune when starting up. This would ensure that the (38MHz-VCO) mixer term would always fall inside the range of the LPF.