Not clear if your problem is the GPS see’s the dip in voltage during starting and switches itself off, or if the starting is pulling the bike battery down enough that power is also being drawn from the GPS battery, ie backewards through the charge cable.
Case 1, this setting may be overideable on the GPS as mentioned above.
Case 2, you could power the socket via a normally closed relay (or the appropriate connections on a change over relay which is more readily available) that is triggered by the starter motor exciter wire. This will have the same effect as manually switching off the the feed to the socket while you start the bike, only it will do it automatically.