I say yes to taking antifungals, but I would try to find other ways to alleviate your skin issues first, if possible. Milk thistle is great for the liver and I can recommend more liver stuff as well. I’d consider trying out artichoke extract which can reduce cholestoral levels but I’ll share my liver detox instead (I’ll make a post about it soon on the forum). I am taking a lot of liver supplements.
I would try out another source of protein than the eggs and see if the hearburn improves (or even reduce the amount you are eating).
I would get some vitamin C for the skin! It works really well.
Garlic is a great antifungal and it alone can cause die-off.
You might want to try able’s ACV regimen first to see if the heartburn subsides or not, and afterward I would reduce or cut out eggs if it doesn’t work.