My humble opinion would be to talk to several dealers (especially any that might be close to you -- moreso if it can be within driving distance) and find a piece that you like a lot. Then appreciate it in person, and finally work towards a price that you can live with. Setting out with a price in mind first is good for the wallet, but may not be as good for satisfaction with what you might end up with.
A signed and papered Muromachi katana in old polish is a possibility. But a new polish alone with most polishers these days runs for a lot more than just 2k. The little to no flaws part is also a big factor in this case; as most lower condition blades of that eras will have flaws from being polished down. For example, I have an attributed Den Oshu Hoju katana in shirasaya in old polish for $2200 but if someone asked me for what you're looking for at that price level, I would politely decline even an effort to search for them. There is just nothing left at that price point to make the search worth the effort.
At the price level you're looking at, you're probably going to have to accept a less than optimal polish or a blade with a fair few flaws or both. Its nothing personal to you, it is just how the market is. I would also probably be acceptive that it is almost certainly going to be in shirasaya, rather than koshirae. Since even minimally appropriate koshirae adds more than 1k value to a blade.
My best advice would be to try to approach I suggested at the start of my post. You should end up with something that you'll enjoy a lot more and have a more solid resale value when you decide to move it on!