I just started trying Body Glide (ingredients list
here). It works really well and goes on easily with a finger, neither of which I was really expecting.

I do notice the
very slightest burning sensation when I put it on, but it's not noticeable during or after a ride, and it's a big advantage that Body Glide doesn't migrate to places where I don't put it. And it lasts a
long time.
I had no idea that Body Glide could be used "down there," but it's right there in the product recommendations, and the ingredients are listed as zero to minimal hazard on the EWG's cosmetics database. (Tribehenin is listed as being a "1" hazard on a scale of 0-10, all the other ingredients are "0.")
I'm still using a dab of Beljum Budder up front, to better lubricate the nooks and crannies.

But like you, I was getting a good bit of post-ride irritation from that, mostly due to it migrating up inside, so I'm really happy with the Body Glide.