What is the nature of conflict between these opposing forces?

I don't know which work by Kipling you are referring to.

Miss Youghal's Sais

There are a few conflicts here. Strickland is British and he is fascinated by Indian culture/people. Both Indians and the British find this very strange bordering on forbidden. Strickland is in love with a woman of greater social stature which is another conflict. Internally Strickland isn't so much at conflict. He seems quite at peace with his odd ways. Internally he struggles at pursuing his love interest but resolves to do so.