Best ideas I had were to grab the JSON that we generate in the Tools menu, and re-order them with that information. This is outside of using the “get code” functionality that we provide that generates some code for you to use in functions.php or so. Just using that would be another way, where you just copy/paste that in the order you want them registered.
Note that CPTUI does not delete post type/taxonomy content. The closest we come is changing the post type slug for the posts, or changing the associated taxonomy. Both of those need user approval. All in all, your content should be safe.
For example, I have this saved data in my local install: https://gist.githubusercontent.com/tw2113/c7e7e165cd4858727262909156f5f1be/raw/65670feca56ae223bd39081dbd59732b01fa9353/gistfile1.txt
However, it’s hard to read, and hard to mentally grep. So, I end up running it through https://jsonlint.com/ to technically validate, but also “beautify” it. That gives me https://gist.githubusercontent.com/tw2113/f22286ed6b4941fdbc02276a091ad7c5/raw/7ab4e6b38faa7f2f6412c936a0270ec1de25e4a9/gistfile1.txt
In that 2nd link, it’s easier to read and understand and you can see that I have 2 post types. “sessions” and “films”. With that in mind, I can literally switch them around, and order them in whatever order I want, like so: https://gist.githubusercontent.com/tw2113/bfd87592ee5aafd5c65019ba91def381/raw/f4476761a4e9a087dc303805ce747ae4b07459ab/gistfile1.txt
Since the import side of the tools menu area takes valid JSON, I can copy that newly ordered version, as is, into that textarea and click import. That will save the exact same post types and settings for it all, without data loss, and it would be in the new order desired.
This won’t be all that efficient in the long run, especially if you’re consistently juggling them around, but it would technically work.