NavList:

Here is another example which sold at auction over 10 years ago for a pretty penny.

https://www.skinnerinc.com/auctions/2447/lots/350

I think you are correct Frank, in that the "bridge" at issue, and separate from the cross piece over the "pillar" which forms part of the sextant frame, has the purpose of stiffening the instrument and protecting the delicate mirror and scope.  In Anita McConnell's book she describes Ramsden's lightweight construction, and so it is a reasonable guess that the addition of this extra bridge may have served a double purpose of stiffening and protection.  Thus they may be considered a variant of the pillar sextants as you suggest.