I know this is not a fix but a work around. I have the same issue using FV and mini or 2017 MBP to eGPU, egpu to monitor. On power up I wait 30-40 seconds then blind type my password. It has worked every time. But yes this should not be the norm.
When I tried a hdmi from the mini to the monitor and a hdmi from the egpu to the monitor it ended up being a nightmare trying to switch between the 2. It was not consistent and had trouble going back to the egpu.
I gave and up pulled out the mini cable, just using egpu and blind password works for me. It is the same on my 2017 MBP as the 2018 mini using the same setup.
When I tried a hdmi from the mini to the monitor and a hdmi from the egpu to the monitor it ended up being a nightmare trying to switch between the 2. It was not consistent and had trouble going back to the egpu.
I gave and up pulled out the mini cable, just using egpu and blind password works for me. It is the same on my 2017 MBP as the 2018 mini using the same setup.