The Fog on the Barrow-downs and The Old Forest quests can be played in Campaign mode and fit into the Lord of the Rings saga campaign. The rules insert for Fog on the Barrow-down doesn't actually mention using a Ring-bearer, though (I don't know about The Old Forest, I don't have that one). If you are playing in campaign mode, it instructs you to follow the campaign mode rules from the Black Riders rules insert. That means using the heroes from your campaign log, which would include your Fellowship sphere Ring-bearer.
At the Fellowship event night, we played without Frodo. We weren't doing a campaign, and nothing directed us to use him, so we just didn't.
Whenever I play it now, though, I use him whether we're doing a campaign or not. The reason being that other saga expansion quests in The Black Riders and The Road Darkens are intended to be played with a Ring-bearer, whether you are in campaign mode or not. Since Fog on the Barrow-downs fits in the campaign, I have been treating it the same way. That means no Spirit Frodo hero, though.
How about others? Do you use a Ring-bearer when you play these PoD quests?