According to the listing in the App Store, the current version of MainStage (version 4.0) requires macOS 15.6 or later.
AI search results claim that Apple originally sold MainStage on physical discs as part of the Logic Studio bundle, and that Apple sold MainStage 2 both on DVD, as part of the Logic Studio (2009) package, and in the App Store.
I was able to confirm that Apple once offered MainStage 2 on DVD; that this wasn't just an AI hallucination.
Apple – Documentation – MainStage 2 – Technical Specifications
The system requirements refer to "Mac OS X v10.4.9 or later", and refer to both PowerPC- and Intel-based Macs. That makes me think that the application is likely 32-bit with no chance of running on Monterey.
Wikipedia claims that "Version 2.1 released in January 2010, introduced a 64-bit mode." Also that "Since version 2.2, updates are available only from the Mac App Store."
https://en.wikipedia.org/wiki/MainStage_(software)
Apple's release notes do not go back as far as versions 1.* and 2.*.
So if the Purchased section of your account in the App Store is not offering you a version of MainStage that works with Monterey, your only hope would be to
- Find an old original copy of MainStage 2.1 on DVD
- Hope that it is compatible with Monterey (since there were many changes other than 32- vs 64-bit which might have caused it to break)
Realistically, it sounds like if you need to take MainStage "on the road", it might be time to buy a new Mac.