I did hear back from Yves at
. I just want to say that every experience I have had with him has been incredible. He responds quickly and thoroughly with real, first-hand knowledge. Priceless!
On the topic of oiling the lattice, he doesn't think that it's necessary because I'll be living in it and keeping it warm with a wood stove all Winter. (I will also have a layer of Tyvek house wrap in the mix so the inside should not be getting wet anyhow.) He also advised something that I hadn't considered - coating all of the exterior wooden parts with an exterior varnish.
Of course! I don't know if I can trust the original paint to be waterproof. I'll look for a varnish that offers UV protection so that the beautiful painted door stays vibrant as long as possible.
I just thought I'd post this all here to make it easy for others to find this answer when they need it.