Orthogonal to eww/gnutls support: in your position, I would (curse a bit and) look into a local https proxy, starting with mitmproxy or tinyproxy. I've been able to use solutions like this in the past, but that past is now distant, and I don't know what's current. I see downstream you ask about some nodeJS options; there's probably a relevant node project that you could set up inside your local network.

Hope that helps,

~Chad