Ooo, I'm very interested to see how the fan art turns out!
I think the most important thing when it comes to faster sketching is comfort with drawing a certain subject. For example, I can much more quickly and easily draw a human body than when I first started drawing. A lot of the time is spent trying to correct errors and stuff.
Your idea of doing timed 10-15 min character sketches sounds like a good one to me, because it'll force you to capture the essential elements of the drawing and you can't get bogged down by the details while doing so. They may look ugly at first, but they'll get better as you get more experience! Though the longer form studies are important as well. (It's like in figure drawing you have gesture drawings and the long analytical figure studies, which are both important on getting better at drawing.)
Also, I have no idea if this video is relevant to your exact issue, but I think it's very thought-provoking and may be helpful, so I'll share it anyway!