I've looked at the National Archive pix for Rd. Nos. 307687 - 696, and none of these agrees with the image of this handled basket shaped posy vase. Unfortunately, for some reason I don't have a pic of 307686, so quickly assumed that logically this basket must therefore be No. 307686 - assuming Fred has interpreted the lozenge details correctly. We do of course know that this basket design is Sowerby pattern 1192 (or 1192.1/2) because it's in the factory pattern books.
However, this basket appears not to be 307686, because according to Cottle that No. has been allocated to a Wall Pocket Vase, with factory pattern No. 1191, and this is substantiated by the factory pattern books. As none of my pix from Kew shows the Wall Pocket Vase, I'm very tempted to assume that 307686 is indeed the Wall Pocket Vase, although why I missed this one I don't know.
So.......... that accounts for all of Fred's Nos. 307686 to 307696, but still leaves us without a Kew image, plus Rd. No., for this handled basket.
However, there seems to be some confusion regarding Rd. 307688 - factory pattern 1187 - with regard to shape and description as detailed by Cottle.
The attached Archives pic shows 307688 (factory pattern 1187) as a small rectangular posy vase with almost identical pattern to our basket - BUT in Simon Cottle's book 307688/1187 is described as a 'basket vase' - i.e. the sort of design we have on this piece Within this situation, might we have an example of the one Rd. No. 307688 covering both the basket shape (factory pattern 1192) and the small rectangular posy (factory pattern 1187)?