I am an Indian citizen. I only have an International Driving Permit (along with a valid us license) with me right now. Will I be allowed to check in for a domestic flight?
A driving licence is a valid ID.
Check this AirIndia webpage, it says
For domestic travel within India, valid photo identification, namely, a passport, Income Tax PAN Card, Voter’s ID or Driving license is mandatory.
So any valid photo identification seems valid. A person on this site has said in this answer that,
I had used a photo credit card without any problems.
A International Driving Permit is a valid proof, but try to take other photo IDs. I know you said you don't have any other ID proofs. But you may use your credit cards / ATM cards(if they have your photo on it), bank passbook, etc...
A photo debit/credit card, Aadhar Card, any Indian State or Government anything where you face can be matched to the photo by authorities would be good.
I would be highly surprised if just the International Driving License is taken as your identity proof as they have no means to know that whether the Driving License is genuine or not.
Make sure that the I.D. you have is not soiled/tore from any place/corner in order to have a pleasant experience.
One hack could be to also use a bus pass to accomplish the same, although it depends on those who check such documentation.
My advice would be to have as much documentation as possible which has your photo identification along with the original and copy of the passport in the US Embassy.That should help them easier to take a call.