This is a very tricky situation given the timeline. AFAIK, if you get a Canadian passport, then you have to rescind your Indian passport. Without that, you cannot get a visa to India. All of this takes time. After Oath, it takes avg 5-7 business dates for Citizenship certificate to arrive, then you will apply for Canadian passport. After that you will rescind Indian Passport and then apply for Visa. Too much to do in such a short frame.
Also somewhere I read in the forum here, that you cannot exit and enter using different passports. It will be flagged in the system.
Regarding the PR card, someone already answered.
I think the only option is to explain the situation and ask for postponing of the oath ceremony.