How to cool a server closet with window?
Question for you to ponder before spending any money:
Do you live in the arctic? Almost certainly you do not, since you just said "you're worried about wasting AC air outside" if you open the closet door.

So with the postulation above, why do you think you can cool the closet with outside air? You cannot. I'd take the stance that there will be many days that the outside air is greater or equal to the 90 degrees you comment on.

Now in the winter time, how will you cool the closet?

The simple answer to your post if you are going to spend the money is to have a window fan, two way unit, or at least one "blower" unit setup, and then you ensure you run some 6" flex tubing from the outlet of the machines to the window, with the blower fan in the piping, anywhere.

Now this said, depending on the house you have, and the mod you are willing to do, you can vent the air from the server into the attic, or into the crawl space, but you still need a way for a cooler source of air to come back into the closet space. Without an equalized pressure differential, no amount of blowing is going to help you, unless you recirculate already hot air.

So..... Buy an AC? Open the window in the winter?
It's generally best to not close off "appliances" that generate heat, in a closed space. Thus, you need to bring cool air in from somewhere, and where it comes from has to be controlled. The outside is not going to suffice for each day of the week and all seasons.