It's an interesting idea, print your own badge maker.

On thingiverse, there is a downloadable set of files to make your own badge maker.

https://www.thingiverse.com/thing:1558914

Looking at the only comment, it may not be the correct scale, but it may be worth investigating, or perhaps modifying?

Please let me know if you do try this? I would love to post a blog article about your success (or otherwise).