The strawberries can be composted, but they will just regrow in there unless you dry them out or really bury them in the heap. If they are really woody in the centre, you might have to chop them up a bit.
Raspberry canes don't really break down unless shredded and if they are full of bindweed, I would just bin or burn the surplus plants. If you are going to move the raspberries to clear them of bindweed, you will have to pick through them really carefully to get every last bit of bindweed root. It might be worth buying fresh stock and starting with clean new canes
