First of all, I never mentioned that I was 'conducting' this research.
We, as an educational institute have the permission to conduct research...
The word "we" is inclusive of the concept of "I". So anywhere you say "we" you're implicitly saying the sentence is logically true if you replace "we" with "I".
Assuming "we" represents myself and Jill:
"We went to the store" is a logical amalgam of:
I went to the store.
Jill went to the store.
Assuming me, Bill, and Ken are all part of the research team
"We have the permission to conduct research" is a logical amalgam of:
I have the permission to conduct research.
Ken has the permission to conduct research.
Bill has the permission to conduct research.
I think that logically demonstrates that you did state you were 'conducting' this research.
Since I was part of this research, I was entitled to say 'our' research, where 'our' refers to me and my colleagues. If you don't think so, then I shall rephrase my statements
I think your repeated invocation of "we" is probably the main source of ambiguity. I don't think it's a grammatical misunderstanding that just requires rephrasing, as you seem to imply. I doubt you could get admitted to any institute except a mental one without being able to distinguish between different sets of "we". I feel certain you knew what you meant when you invoked "we" each time. You mention "entitlement" to use the first person plural, but personal pronouns aren't a matter of entitlement, it's a matter of logical description, as I demonstrated above. I don't think the issue was challenging your entitlement to use the word "our" or anything so semantic, the existence of the research and project were being challenged. We could sort out whether or not the research counts as yours once we establish its existence.