The editor can export a text version of the preset settings. XML as well.
I don't think there's a corresponding import, so I don't think that will help. Even if there was an import, it's using ascii instead of base64, so that's probably not a good solution.
I wrote an app to do the preset encoding/decoding with drag and drop to make the process as streamlined as possible. This works much better than saving a preset .syx file, but it's still awkward. There are 4 steps to save a preset into a track note and 3 steps to restore it from a track note. Also, this doesn't work in all DAWs. For example, Cubase can't store a note that large.
This illustrates the point that a plugin with a one click save and one click restore would be a much simpler and more universal solution.
Edit:
For anybody who wants to try it, here's a google drive download link for the Mac app. Put it in the dock after you download and unzip it. (No warranties, use at your own risk, etc.)
Code:
https://drive.google.com/file/d/1jutYVpLPpJKxbx63QCfxqcvYeY_2NRl-/view?usp=share_link
Instructions:
Save:
1) Export the preset in Axe-Edit to a .syx
2) Navigate to the folder in the Finder where you stored the .syx
3) Drag the .syx file to the encoder app in the dock. This encodes the preset, saves that encoding into a .b64 file, and puts the base64 text in the clipboard.
4) Go to your DAW and paste into your track note. It'll be quite large, at least 60KB.
Restore:
1) Copy the preset text from your DAW track note that you pasted earlier.
2) Click on the app icon in the dock. This will decode the base64 clipboard text into a axefxpreset.syx file in your home folder.
3) Use Axe-Edit to import that axefxpreset.syx file in your home folder that was created in step 2.