Since it works for me with stronger adapters but not with the original Apple adapter for the iPad Pro, it can only be due to the power of the adapter. The LMP Travel Dock also needs power.

This is due to the charging adapter from the iPad Pro. The Apple adapter from the iPad Pro refuses to give power to the LMP Travel Dock. I.e. it briefly gives power to see what is connected and then stops the power supply shortly afterwards. This is also the case with me. Presumably the LMP Travel Dock requires a power class that the iPad Pro charging adapter does not support. I assume that together the iPad Pro and the LMP Travel Dock require more power than the iPad Pro charging adapter can deliver. But with the roline 6-port charging adapter or the Apple 61W USB-C power supply it works.