Thanks, Chris.
Neither tapping menu button nor holding “1” for 3 seconds worked.
Today for some reason, the Sidekick did allow me to do MENU → 5124 {STOP} and the 5124 played back (unlike previous recent attempts, but like it did during initial setup).
Still, via an API call the mode is still 0, keys is still 0, the chans is 5, techs looks to have an array, chan_links looks to have an array, key_links is null, firmware is “null”, and the keystream attribute seemed null-ish:
{"__modified":"1675529808","name":"SHxxxxxxxx_Wall","location":"Virtual","mode":0,"keys":0,"chans":5,"techs":[240,240,240,240,240],"chan_links":{"1":[],"2":[],"3":[],"4":[],"5":[]},"key_links":{},"signal":100,"battery":100,"model":"SKS-500","fw_ver":null,"_":"230c8d58","__":"8a1d5e09","keystream":{"_":"00000000"}}
Verified BPUP was still not sending keystream events.
So via the Bond app, I did the remove Sidekick, then re-added (including holding the “1” button on the same Sidekick).
Now the Bond app shows, for that Sidekick: Mode unsupported – your Sidekick is set to 8 scene keys mode, which is not supported on the app for now, but it will be in the near future. But rest assured that you can still use it with third-party integrations.
BPUP keystream events now seem to be being sent without issue. Hurray! 
And now an API call shows what I assume to be a correct mode of 8, keys of 8, the chans is now 0, techs now has an empty array, chan_links now has an empty array, key_links is now populated with an array of 8, firmware is still “null”, and the keystream attribute seems less null:
{"__modified":"1677764796","name":"SHxxxxxxxx_Wall","location":"Virtual","mode":8,"keys":8,"chans":0,"techs":[],"chan_links":{},"key_links":{"1":[],"2":[],"3":[],"4":[],"5":[],"6":[],"7":[],"8":[]},"signal":100,"battery":100,"model":"SKS-500","fw_ver":null,"_":"4a127e33","__":"cfab7a90","keystream":{"_":"130302d0"}}
I did leave the other Sidekick in the “broken” state, in case you want to do any additional troubleshooting.