I think you are misunderstanding what a regressive tax is. A tax can be flat AND regressive (in fact, those are usually the kind that are). It is about the BURDEN of the tax being unequal, not the dollar amount.
For example, a speeding ticket is a regressive system, because a $200 dollar ticket is a lot more damaging to a poor person than a rich person.
By "flat" I mean a fixed percent of your income. A fixed $200 isn't a flat tax, because it's not a fixed percent of your income.
Yes, apparently-flat taxes can sometimes be regressive, for example a flat sales tax can be somewhat regressive because poor people spend a higher percent of their income compared to rich people. But this thread isn't about sales tax, so I don't think the flat tax in this situation is regressive.
For example, a speeding ticket is a regressive system, because a $200 dollar ticket is a lot more damaging to a poor person than a rich person.