As you already found out, there is no direct way of finding download numbers for a specific app.

Some companies disclose such information when milestones are reached or within interviews and blog posts. Did you try searching?.

If you're just interested in how competitive a category is, for ASO purposes, you can checkout app worth per category in SensorTower. I doubt the actual figures are accurate but it will at least give you an idea of how they stack against each other.

Alternatively, you can look for 3rd party articles like this: http://techcrunch.com/2013/06/27/how-do-you-break-into-iphone-app-store-top-50-try-23k-free-daily-downloads-950-paid-or-12k-in-daily-revenue/

Let me know if you want more help with you specific case.