Good fortune today.
He ummed and ahhed for a bit until I mentioned pulling it, then jumped at the chance since he was doubtful any repair job would be worth the effort.
After 30 minutes of strain and perspiring from them (dentist and trainee) and litres of tears from me they managed to get it out neatly in one piece, a job very well done which has impressed me.
No more toothpick for a while, and hopefully very little pain now.
The downside is they still plan on scaling them on my next visit, pity as I'd be happier if they just pulled or patched.