What FAA certificate do you have that required you to submit a TSA request in the past (excluding a FAA 61.75 based on certificate)?
The school is not SEVIS approved because they have elected to not go and get Part 141 status which costs many $$. As a result they cannot sponsor a foreign student for a M1 visa and therefore will tell you a story that you don't need a visa in order to get your business - which is clearly incorrect as you say you are doing flight training towards a FAA IR attached to your 'based on' certificate.
You will also need to submit a TSA application for the IR training and meet the training requirements detailed in FAR 61.
Where are you planning to do your IR Flight Test when the time comes?
At the end of the day, you are responsible for your visa status not the Flight School regardless of what they might say to you.