It is good the measure the impedance in balanced setup, but that can be done with a single port and a balun also (so you can do a single port measurement).
As you are interested in a narrow frequency range, balun functions can be constructed in various ways.
Some examples you can find here:
https://www.edaboard.com/threads/248079/ and in many other places.
Calibration with balun is the best way, but the quality of the calibration depends on how good you can make the open, short en load. If you use very thin cable, all parasitic effects are less. If you have a very good balun (and you know that for sure), you may leave it out of the calibration, and do manual correction afterwards (electrical length, phase, amplitude). Even when calibrating with balun, you may need some manual corrections (because of bad short open and load), depending on what impedance you are measuring.
When measuring the antenna's impedance, the balun should be in the H-plane of the antenna, otherwise still common mode current is induced and these can render your results useless.