Disclaimer: I've not used any Canon Wireless File Transmitter with either my Canon 5D Mark III, 5D Mark IV, or 7D Mark II. I'm also in the United States where the radios use version A of Canon's wireless products to comply with radio frequency emission regulations. The only difference between A, B, C, D, etc. versions of Canon's radio emitting products is the region for which the internal radio is compliant.
I have read multiple statements in the Q&A and review sections of sellers such as B&H, amazon, Adorama, etc. where other users claim to be using the WFT-E7A (ver 2) with 5D Mark III cameras. From the Q&A section for B&H's listing of the Canon WFT-E7A:
Other than the updated firmware needed for compatibility with newer camera models, it seems to me the only real difference between the first version and the (ver 2) is the inclusion of two additional cables (one long and one short) with an USB 3.0 Micro B connector to allow the WFT to be connected to a 5D Mark IV or other camera (5Ds, 5Ds R, 7D Mark II, for example) with the USB 3.0 Micro B socket. The 5D Mark III has an USB Mini B socket.
Older examples of the WFT-E7 need firmware updates to work with cameras newer than when the specific WFT unit in question was shipped. The firmware may be updated by the end user by attaching the WFT unit to a compatible computer and using the EOS Utility application.
The first version of the WFT-7A may be used with newer cameras using other connectors by updating the firmware and using a correct cable, such as the Canon IFC-150AB II USB Interface Cable for WFT-E7A Wireless Transmitter. The notice from Canon regarding updating an older WFT-E7A to work with the 5D Mark IV is at this link. Support for the EOS R was added with the next firmware version announced here.