First, it is important to understand why the movement of the change should be done without any acceleration.
This is because, in order to accelerate the charge, the force applied to it has to exceed the force of the field acting on the charge and, as a result, the total work done on the charge, while moving it from the reference point A to some point B, will exceed the potential difference between A and B, by the amount of the extra work spent to accelerate the charge.
Once that understood, you should see that the suggestions in the comment - to move the charge very slow - will lead to as accurate measurement as you wish, given that, in theory, you can move the charge infinitely slow.
If, however, you want to complete the measurements quickly, there are a couple of methods which would still give you accurate results.
1) You can accelerate the charge from the initial zero velocity at point A to any desired velocity and slow it down to zero velocity, when you arrive to point B, in which case the positive work spent to accelerate the charge will be canceled by the negative work spent to slow it down and, therefore, the total work will be exactly equal to the potential difference between A and B.
2) You can accelerate the charge from the initial zero velocity at point A to any desired velocity, move the charge at that velocity all the way to B and subtract the kinetic energy of the charge from the total work. The result will be exactly equal to the potential difference between A and B.
3) You can accelerate the charge to a desired velocity before you arrive to point A and then continue moving the charge all the way to point B at that velocity. In this case the total work, performed on the charge while moving it, at a constant velocity, from A to B, will be exactly equal to the potential difference between A and B, without any adjustments.
The advantage of method 3 over 1 and 2 is that you can accurately measure the potential relative to A at any point on the way from A to B and end up with an accurate potential map of the whole path. You'll have the same ability, if you move at infinitely low speed, but it would take more time.