Which bit of the Linears is is that doesn't like the DC?

The transformer.

Would running the inverter into an isolating transformer (shaver
socket) and connecting the shaver to the output of that be any better?

No.

The problem is dc just make the transformer heat up, with no (easy)
solution available.

However I'd be surprised if recently manufactured rechargable
shavers use linear supplies, when everything else is using switchers.