Would be interesting to try open the gates, allow them to start moving and then ramp up the current on the Tesla and see if they still stop.
Like others have said the voltage drop of the cable run could be the issue and depending on the CSA (cross-sectional area) of the cable then the voltage drop can increase. I have seen this with a 20M extension (Tough Leads) when using the Tesla granny charger. Tesla states it's receiving 224V when using the extension, and around 236V without the extension.
I would assume it's 1.5mm2 3-core SWA they have buried in the ground for the gates. In which case based off your current draw estimate it shouldn't be dropped more than 5V. If it's a smaller CSA then this will increase.
Not sure what gate controller you have, but if it has an older supply then has a static AC-DC converter for the control electronics, then it might not be regulating the DC output well enough. I have seen some PLCs (programmable logic controllers) from Siemens that have a 24V DC input but if it drops down to 19V it will just switch off.
It is odd though that the BMW isn't causing the same issue and would make me think it could be the Tesla putting some DC transient voltage back onto the AC supply. Especially considering that with both chargers you are experiencing the same issue.
One last thing, has this started since it got colder or after heavy rain?