That sounds like a tough one. When a person in a leadership position is not supportive there may not be a lot of options. Being unfamiliar with the situation I’ll make a few suggestions to see what helps. One idea is to understand exactly what the project leader believes will help. Is there any data collection going on at all? What does this person believe is valuable for this project?
Another idea is to find easier ways to collect the data. And lastly, have you made a case for the value of the data collection? Have you described the potential benefit? If the team collects the data and it helps point to useful solutions, would that benefit the team and other stakeholders? Making that clear may help your effort. Good luck!