Interestingly, the away team has a slight advantage overall in extra innings games since the inception of the ghost runner.
From 1901 to 2019, the home team winning percentage in extra innings games was .523.
From 2020 to 2024, the home team winning percentage in extra innings games was .493.
If the away team can score in the top of the inning, even one run, they have a decent chance of winning the game. For that reason, I agree with your opinion of bunting the runner over to third base, in general.
That said, it really depends a lot on who is at the plate and who is on deck. If Anthony is up with Bregman on deck, I'm swinging away. If Wong is up with Anthony on deck, I'd probably bunt. This also assumes that the guys can actually get a good bunt down.