I was refused tech help and referred to supportforums. I found a tip there that (at least for the moment) broke the logjam and now my RV340 can send out g-mail. It was remarkably Byzantine, to accomplish such a routine, dull objective. No help from Cisco itself, whatsoever. I surely didn't want to go the RMA route, as I knew the next router would fail the same way...

At least I owe you thanks for pointing me towards an eventual solution.