Is core insulation sensible with a 3 cm air gap?

Hello everyone,

is it sensible to insulate a cavity wall afterwards if the air gap is only 3cm? The clinker bricks are also connected to the masonry (pumice) with steel pins. Whether mortar also creates a connection somewhere is unknown. The air gap is still, at least I don't see any air slots at the bottom. And at the top, the concrete ceiling sits directly on the clinker.

My question is, if insulation is sensible, which material is best suited? Aerogel is out due to the immense additional cost. Definitely something that hardens and does not crumble.
And, are cold bridges possible due to the steel pins? If yes, is it serious?

Thanks and regards

The steel pins do not create a cold bridge. There is an insulation material made from recycled paper that is blown in. I am not allowed to mention product names here. In any case, they are soaked flakes made from recycled paper and also fill tight gaps as they are blown in.