Hi Gus
Like it or not, those fruit flies (and other insects and crawlies, and their larvae) are actually working for you to break down your compost. As long as it isn't something that can sting or bite you (bees, ants), can you work around them? Most of us now have masks laying about, you can slip a mask on when you are making a "deposit".
Next to my compost pile, I keep a covered bin with chopped leaves and grass in it (my chopped mixed greens and browns to add in). When I place fresh kitchen material on my pile, I also toss some of the greens and browns over the fresh stuff, it doesn't eliminate flies, but it keeps the numbers manageable. This should work in your wheelie bin too.
I also agree with checking that drainage as Yorkie said.