I would call this a premature conclusion.
There's a whole variety of different microphone types for different uses around. Unfortunately neither did you specify which microphone(s) you have tried, nor could I find any specification in the technical data ("audio" section) of your camera which type of microphone your camera requires.
Note however, that Sony does offer several microphone adapters to connect external microphones to your camera, which are "sold seperately".:
Just because a plug is mechanically compatible, i.e. 3.5mm minijack, doesn't guarantee that plug and socket are electronically compatible as well. For instance, 3.5mm jacks come in varieties of 2, 3, 4 or even 5 contact surfaces. See this for more details.
How true. Maybe you can still go back to your dealer and ask for advice.
An utterly cheap way to try, if this theory holds any water would be, to attach the camera directly to the gimbal, without the grip. You would very quickly find out, if this gimbal lives up to its touted performance. The external battery pack would then just add the additional operation time to the whole combination, if you find that the gimbal itself is worth the effort and extra expense.
(I always try to keep financial risk low).