I would contact each of the manufactures directly and ask them specifically in writing what you have posted here.
You have answered the question for yourself, but your budget is a bit on the low side.
Always a compromise between precision and what your willing to pay to get it.
The machine with ballscrews is going to win.
Unless you can set up backlash compensation real tight and the computer can effectively take it out. But then your going to keep measuring that, you can't just set it and forget it.
E-mail your requirements to each manufacture and post here your response.