At first glance, I am wondering "why"?
I have a macro running that turns off both DATA, and WiFi when the screen is locked. I do that on purpose to preserve the battery.
IMO, I do not need to know that someone on FB, or Messenger has left me a message at 03:dark thirty.... so, I make that wait until I unlock the screen.
Is there a negative? yes, I don't receive Storm Warnings until long after the fact when I forget to turn the macro off on stormy days.
So, with my experience with this, I am going to suggest that you investigate if the "Internet" is being turned off somehow?
Something obviously is wrong.
Investigate the priority of the notifications for the app involved.
There is a possibility that "Notifications is off for that app".
Google Calendar works on my phone, even though the Internet is turned off, so they are maybe using the SMS function to make that happen... I am going to suggest, that this is the case.
( more post thought on this: nope, it is from the Google Calendar app itself that is installed on the phone, it sync's up with the Internet, and therefore, the app knows when to alert you. So, it needs to have Priority over the Screen Lock function. )
I have several alarms set to tell me to carry the trash bin to the street, to get on the Tread Mill and exercise my legs, etc... all of them work, and all of them come from Google Calendar.