This problem could be caused by instability in the transmitters amplifier. If the transmitter in question is from BW, then they are known for instability. Particularly the V3 versions. Instability is when the amplifier stage oscillates by itself. That will usually only happen when the load is narrowband which can the the antenna, bandpass filter or both. A spectrum analyser is good to have to diagnose this. Typically what you see is a pair of carriers either side of the main frequency spaced 1-3MHz and of the same amplitude. They cause reflected power because they reflect off of the narrowband load.
You can diagnose instability without a spectrum analyser. Turn down the transmitter power and you will find that the reflected power drops suddenly at some point. On high power transmitters it is often mistaken for arcing or insulation breakdown in the antenna system. There is an easy fix which usually works and it does not involve buying a better transmitter! Change the length of the coaxial cable connected to the transmitter output by a few feet (try 3 feet). It can be longer or shorter. It may take a few attempts to find the right length.
And the comment from Cold Coffee about how close the antennas need to be to each other is spot on. They do need to be very close to get enough power transferred to cause the transmitter to shut down.