Refinishing stained wooden doors
Obviously, I can't simply stain over the existing stain

I dunno why not. The simple solution to making the wood a darker stain is to add more coats and/or use a darker stain. It's quite common to apply more than one coat, mix stains to get the color you're after or apply different colors in sequence. Most of the wood trim in my house has multiple coats in multiple colors, and some has been applied with many years between the trim being applied, so the color match was simply done by eye.
If you have some scraps, do some testing on those as follows:

Apply one coat of the same stain, using the same technique as was used on the existing woodwork. This will give you a baseline that's as close as possible to what you currently have.
Apply a second coat of the same stuff to a small portion of your sample.
Apply a heavy second coat of the same stuff to a small portion of your sample.
Apply a slightly darker stain to another small portion.
Apply a much darker stain to yet another small portion.

Rinse. Lather. Repeat. Until you have the color you're after. Replicate those steps on the rest of the trim to replicate the results.
You may end up having to buy 3 or 4 different color of stains to try things out until you finally wind up with what you want. Just make sure you keep detailed notes on each sample section so you know exactly what you did so you can reproduce it when you find what you're after.