What you describe is not uncommon. You certainly want to keep the look of your door inline with the style of the home but wood doors will always have some challenges in certain exposures.
Ten years ago, I would have suggested you stay with a wood door but plan on re-finishing this pretty frequently.
Fortunately the technology of Fiberglass doors has advanced to the point where I would strongly recommend this product.
They are available in many traditional styles, most manufacturers can provide custom size doors and they will continue to look as good as the day they were installed with only a yearly cleaning.
The budget you shared may limit you somewhat, but you should be able to find something very nice about that price point. Check into it with a good General Remodeler who can help you to source a door in your area. I have had great results with Therma-tru and Pro-Via doors here in Ohio.