by BGS, do you also include side-mounted (as opposed to overhead-mounted) green signs?
for example, is this a BGS? it's B, and it's G, and it's certainly an S, but it might be a borderline example given its purpose.

a quick survey of the shield gallery reveals, offhand, only ID, KY, MO, ND, OH, OR, SD, TN, WV with no state-named shields on green signs in photos taken at any time, including some examples from only the 60s (RI), but most with examples surviving into the early 2000s at least. ID, OH, OR, SD and TN are likely the results of insufficient data, as they got rid of state-named shields quite a while back. MO, ND and WV are very scrupulous in their standards compliance, and KY is a little bit of both, I think.
(LA would be an example as well, but there is a brand-new state-named I-10 shield on a green sign which I have not yet posted to the gallery.)
restricting it to signs which are known to survive to the present day gives a few more negative examples, but those are directly in proportion to states in general not using state-named shields (Delaware, Utah, etc).