However, you wouldn't have a possibility to "log in" to a container or interfere with it in any deep way, because it would break the concept.
I think that the "serverless" concept already treats the running service as an immutable thing so probably adding the proof of running from a specific checkout would be easier there.
https://www.gnu.org/philosophy/who-does-that-server-really-s...
For example, many OSS SaaS products I've seen have an open source version that only supports running a single user or single company but their deployed SaaS version is multi tenant.
So if you're really concerned about SaaS products slipping in back doors or something like that that defeats the "publicly audit-able" condition (if I'm reading your concern right) you can't do anything about it except run the OSS code yourself and not use the SaaS.