OK, so first of all, I cannot spell, i.e., should be "Persistent". Second of all, Sanjana Goli sent a very nice explanation for what I was doing wrong, which mainly consisted of using the Function Keys in combination with Ctrl and so forth. This may work for some combinations, but I suspect that, even if Fusion does not throw an exception during the assignment in the Sketch module, for instance, the combination may exist in another module, say Electronics Design. The following is the response I received in total. Particularly helpful is the link to the official Hotlinks Guide.
Hi Colin,
Thank you for contacting Autodesk Technical Support.
My name is Sanjana Goli. This support request has been escalated to me for further review and I’ll be assisting you with this.
From the case description, I understand that your custom hotkey assignment for Combine using "CTRL+F10" is not stored in UI between sessions.
For successful hotkey assignments in Fusion 360, please use any of the following combinations:
- Shift + Alphanumeric key
- Control + Alphanumeric key
- Alt + Alphanumeric key
- Function key
- Alphabets except for reserved hotkeys
Please check if this works on your end as I've tested it on my end and it seems to work fine. The hotkey assignment is also retained between sessions. Once, the hotkey is assigned please ensure the change is committed.

Please refer to "Fusion 360 shortcuts & hotkey guide" for information on reserved hotkeys.
I hope this helps you in resolving the problem. Please feel free to write back to me if you need any further help with this. I will be more than happy to assist you further in this.
Thank you again for contacting Autodesk and have a nice day!
Best regards,
Sanjana Goli
Global Product support