This kind of integration testing is best left up to the individual distros. Same as the integration (as in: packaging) itself.
Distros don’t want your binary package, they want your source code, build instructions and a build system that won’t make them cry. Some distros even explicitly disallow re-packaging external binary distributions.
As a distro maintainer, I appreciate your wish to do QA on all the distros but that’s just too much work. You focus on making your software better, we focus on making it work with the rest of the software ecosystem.
Providing a package for one or two distros (i.e. your favourite one) is good practice to ensure your software can be reasonably packaged but it’s not the primary way your users should receive your package in the traditional Linux distro model.
Additionally, you might want to package your software for one of the cross-distro package managers such as Flatpak, AppImage, Snap, Nix, Guix, distri or homebrew. This can serve distro maintainers as a point of reference; showing how it is intended to work so they can compare their packaging effort. If there’s some bug present in the distro package but not the cross-distro package, that’s a good sign the issue lies in the distro packaging for example.
Again, don’t put much time in this. Focus on your app.