It would be nice to have a registration document AND normal online registration for the same class...

This is not really possible. However, if you want to get adventurous, there is a work-around, but not for the faint of heart!...
Set up a class with the registration document you want. Before leaving the page where you set the registration option, right click (Mac ctrl click) on the link to it that appears below the text box and "Copy link" or "Copy Shortcut". Then pick the registration with message option and put your message text in along with the following html reference to the document

....... <a target="_blank" href="paste the link you copied here"> registration document </a> .......

If the pasted link shows as https://, change it to http:// (no s). The words "registration document" will become a link that opens up the document you set up earlier. HTML is very picky, so be sure to test it out after you have saved the class info. Good Luck!
Gerry