An enlarged island is definitely needed. I'm not sure about restricting turning at the junction as I'm sure it would annoy somebody and lead to unintended consequences.
As an example, if you prevent straight on traffic out of Dartmouth Road, then Manor Mount is likely to experience more traffic from cars using this to get back onto the east bound south circular.

What would be interesting to find out is where cars using Dartmouth Road are traveling to and from. Except for local people I can't imagine it is a particularly useful route for most people. It is not the obvious route to the South circular from Sydenham or Crystal Palace unless you are heading for the centre of Forest Hill.

Perhaps traffic on Dartmouth Road could be reduced by diverting the A2216. This A road starts in Sydenham (Cobbs Corner) and goes down Dartmouth Road, later is emerges on Lordship Lane and continues to Denmark Hill. In other words London Road is both the A205 and the A2216. A faster route is obviously to use Sydenham Hill and cut out the centre of Forest Hill. Recognising this as an appropriate route for traffic on the A2216 would sort out any problems with Sat Nav systems which prefer A roads to un-numbered roads.

It would also help traffic from Catford realise that the best route to Crystal Palace is up Sydenham High Street (with its excellent pedestrian crossings) or through Penge, rather than using Dartmouth Road and Jews Walk.