If nothing works to your satisfaction you have 2 choices.
One is paint it. Quick, easy, and cost effective.
Second, go to a paint/stain mixing store and take them a piece of pine and one of your stained doors. Have them custom mix a stain that will go on the pine that will match your stained oak door. It will cost about 20-25 bucks.
Odds are the plan you have in place will work out great. But there are options if it doesn't.