Thank you for your response!
That’s a good idea, but unfortunately complete pooling is not possible in my real world dataset for domain-specific reasons. To attempt to extend the toy example above the test scores cannot be pooled directly because school A gave a math test and school B gave an English test. The statistic I want is a measure of how much tutoring affects either math or English grades across schools.
It sounds like I have a solution: a weighted sum of the individual school treatment effect parameters. I have already tried this in my real world dataset and it gives some expected positive results, thank you for confirming this is acceptable!