If it was me, I WOULD definitely move them (be extra careful if you do - they are delicate).
Before I mowed my lawn l "swept" the area that I thought they'd be in with my hand - this made them visible - they hop like crazy to get away. They can be encouraged to go into the border by continuing this sweeping action to "herd" them that towards the edge of the lawn.
I don't think you can be expected to save them all. I wouldn't prolong mowing - I would imagine the accidental killing of a frog gets more gruesome the larger it has grown. And letting the grass grow will mean they all want to go into it!
