As you are concerned for the time being with the NEET councelling, I would say that for the time being, you can make an affidavit and keep it ready as you need not want to waste time running behind it but prepare yourself for NEET and give your best try. During the verification, you can produce the affidavit and clear the ongoing scenario. Please keep in mind that you always provide single or same formate for all your future reference. I mean always use the same format that is given in the Xth or Xiith standard certificates. Making changes in your mark sheets and other papers are tedious and time-consuming.
On the other hand, you can either visit any nearby Aadhar center and make an appointment for change in name and ask them to change the middle name i.e. of your father to initial as in your certificates or mark sheets. Once your Aadhar card changes are made, do take the print and along with your certificate do apply for changes in your domicile certificate. These will take only a day or two for a manual approach and will take 7 - 14 working days to get the domicile certificate with correction. In this way, you can make a common format for all your documents and always keep in mind to follow the same.
“The most important thing in life is to learn how to give out love, and to let it come in." — Morrie Schwartz