If you upgrade to 13.1 you are going to need to:
-Set up new phone as a new device
-upgrade the phone to 13.1 (if available day 1 or download the beta profile or restore to an earlier version)
-erase all contents of the phone
-restore from 13.1 back up
I am on the 13 beta and apple had to replace my phone last week. I had to do the above process. Set up as new, download the beta profile to upgrade the phone, erase the phone and restore from my 13 backup. It may sound like a headache, but it took ~10 mins and did it all on the apple store wifi.
I do this every year when new phones come out. While it's true that we have never seen a .1 beta release before the GM of the .0 , we do usually see something like a .01 beta between the GM and launch day for the new phones.
That's why I developed this process to set up my new phones.
Let's look at 2 years ago with the iPhone X, GA is released on keynote day-I upgraded my phone. A few days latter they released a .01 beta and I upgraded to it.
It's launch day and I am excited to get my X. So I have UPS hold my phone at the ship center and I am there at 7:30AM to get it. I'm out of there by 8 ish with the new phone and rush home thinking I'll have it set up and running before lunch. Wrong. I forgot that I have that .01 on my old phone and it has backed up. I spend 3 hours trying to find a way around this situation and be able to set up the new phone and restore my backup. You see there is no 11.01 for the X. just as my frustration is at it's peek- they release a 11.01 for the X on the developer portal. I quickly upgrade to it and restore.
Many beta users (including me) will be happy with their 13.1 for the next 23 days. The first part of that 24th day is going to be a challenge. But I expect that Apple will help us, just like they have done in the past.