jon wrote:
I'm aware that a piece of wood (I forget its name) can be used to
prevent the break out as a router bit exists its cut from ruining the
work piece.

Sacrificial strip. Its purpose is to prevent breakout when routing across
the grain on natural timber.

I have a "breakfast bar" that needs finishing using a
router. This has both edges finished as a curve making the use of a
sacrificial piece of wood difficult/impossible.

Whats it made of? And in which plane is the curve? Do you mean both
corners are a radius?

Why does it need finishing with a router specifically?

Anyone know if
routing a worktop like this will cause problems or does chipboard
behave differently anyway?

Chipboard doesn't have a 'grain' so it machines the same way in any
direction. There is no grain to break out.