I will comment on this, not with any solution but hopefully it helps.
There is an APNS "budget" that they talk about in this doc: https://developer.apple.com/documentation/activitykit/updating-live-activities-with-activitykit-push-notifications#
There is quite a bit of information which we have utilized with limited success (because it is so difficult to test).
Primarily using more priority 5 pushes, and setting the frequent live activity updates flag in my Info.plist.
Good luck and let me know what works for you!
Topic:
App & System Services
SubTopic:
Notifications
Tags: