I guess there's no point in pointing you to Inns that serve meals without lodging then. (Like this one for meals between 3pm and 8pm http://www.fukuzumi-ro.com/english/e-rate1.htm)

Hopefully someone can give you personal recommendations.

General dinner advice for tourists is at the places listed here:
http://www.odakyu.jp/english/lnk/index.html

I can reassure you that at lunch time there are many more places to choose from, especially in this area
http://www.hakone.or.jp/english/youkoso/mapg-e.html

As you have no doubt discovered, there's not much in English about dining options in this area. Although I don't usually rely on hotels for recommendations, I'd probably make an exception in this case. Avoiding French food and Kaiseki menus at dinner time might be a challenge.