As I said earlier, the example would be Surtsee. Those massive effects are from already established volcanoes going bang. Getting a Surtsee style volcano in the Straits would block them over a period of time and wouldn't necessarily put huge amounts of gunk into the atmosphere.
The one currently playing around in Iceland is creating the problems that it is because of the ice and snow coverage on the surface, take that away and you don't get the ash cloud.
And Surtesy, at it's absolute maximum cover around one square mile. The Strait is 13km (7+ MILES
wide) Suttesy is also in a geologically active region, something that the Straits hasn't been in a half billion years or so.
Of course you get ash. Look at the eruptions of Mt. Saint Helen's, Mt. Pinatubo, Mt. Vesuvius, Krakatoa, or the historic record for Tambor, Tombo, or any of the mega-volvanoes of lore. For that matter look at the Surtsey site. About the only place you don't get serious ash is in the very rare cases such as Kilauea, which is a rather unique case, one that would have resulted in a SERIES of volcanoes and almost unquestionably no Strait to begin with.
This is a total ASB idea and needs to be treated as such.