If the update is accepted, then everything works fine.
I'm not sure what do you mean with "accepted". There's no option that accepts or performs an update. The closest to an "accept", doesn't do anything, it only display a message asking to quite the sim and run the FSDT Live Update from the Desktop.
The other two options shows the release notes, and Ignore the update. Both are working fine and no keyboard/mouse is affected.
In addition to that, please note that what you are looking at, it's not a "GSX menu", or an "FSDT menu", it's the standard Simconnect menu that is used by all add-ons, and mouse and keyboard handling is done by the sim entirely outside our control.
Since it doesn't happen here, I can only guess there must be some interference from one of your other add-ons, possibly one using that menu, since there's only one Simconnect menu, and it's shared by all add-ons. The first obvious test is trying if this happens again with everything else disabled except GSX.