You might be confused about what "beta" means in this context. In spite of being a software related site, this has nothing to do with the software powering the site. Unlike the release process for new versions of software where pre-release test versions are known as "betas", the term "beta" in the context of SE sites refers to the process of setting up the community and scoping out the overall subject matter of the site, not the software. The beta phase is when people that were interested in the topic get together and demonstrate what questions can and cannot be asked inside the scope of the site. This is a phase that sites go through once, then eventually graduate. No matter how many iterations of the software happen, the process of picking a subject matter for a site and building up the initial community is only done once.
Once the beta period is over, it's never coming back. StackExchange is not a time machine. A limited number of badges could only be earned if you were in the right place at the right time. They were on-time-shots. If you didn't participate in the beta stage, you don't get a beta badge. End of story.
You could however earn a beta badge for another site by participating on Area 51 and finding a proposal you are interested in that is going into beta.