[Merry]

It's too early for a civil certificate (1837+) but you could get a copy of the Parish Register entry from the Hampshire RO, for a fee!

Otherwise, you should also check the Chaplains Returns marriages (overseas ones and I think they may include some UK army marriages as well) if you haven't already (they are on FMP 1796-1880, but you'll need a sub) and bear in mind both their surnames are open to being listed under different letters of the alphabet!!

I'm not sure about the permission bit. Certainly at various times one needed permission from a commanding officer to marry, but I think sometimes it made a difference what rank you were and at what time in history you were marrying. Plus of course, needing permission didn't mean that permission was always sought!