How essential are the eggs in this recipe? I’m allergic but I love all the flavors in this recipe. Doesn’t seem like a typical egg replacer would work

Another reliable idea...use same volume of silken tofu to replace the eggs. Puree first, then add to mixture.
On fat content...tofu has approx 10% compared to eggs approx 15%, so add 1-2 tablespoons fat to filling.

I'm currently in the midst of my own eggless, custard-based pie recipe struggles, so maybe don't trust me BUT I think the replacements I've seen, like cornstarch instead of eggs would work here, at least for thickening. Replacing the richness that comes from egg yolks, I'm not so sure about.