Ballet flats by their design is that they have minimal design features in them as there is just not enough room. This poses a problem if you want to wear shoes like ballet flats and need or want some arch support. There are no ballet flats that you can buy that have an arch support built into them.
Podiatrists often face a challenge in a number of circumstances in which they need to get foot orthotics or support in shoes when there is not enough room in the shoe. As well as ballet flats, there are football boots and cycling shoes that pose a similar problem.
Often the best you can do is use some simple felt or silicone padding and fashion some sort of arch support or foot orthotic. That will be less then ideal and better than nothing. Often clinical practice is one of compromises and unless you are willing to change the shoes and not use ballet flats, then this is one of those compromises.