Same problem, but with iPad Pro 9.7 (2016). Started after 11.2.1 update. First, the badge app icon (red bubble with # in it on upper fight corner of app icon) froze at "2" notifications (it was obvious that I had no new messages). Called Apple Support. They escalated it to a second tier rep, but he was perplexed too. He sent a message to the engineers and made an appointment with me to check back in a few days. I wondered the heck would happen in those few days and thought, well, nothing.
So I tried to troubleshoot at home when all my devices (iPad, iPhone, iWatch) were on one WiFi. I erased everything in Messages app on iPad, but still had the "2" frozen on the badge app notification. Then, I thought it might be an iCloud issue or network issue so I logged out of messages on iPad and iPhone, turned off and turned back on all settings in notifications for Message app, changed network settings, rebooted all three devices, logged back into my WiFi all three devices, logged back into Message app on iPad and iPhone, and just in case, turned off and back on all notifications settings on iPad and iPhone. Then, tested iPad notifications with sending text from iPhone and NOTHING. iPad still stuck at "2" on the badge app icon while the iPhone updated the badge app icon with a "1" as normal.
What seemed to work (at first) was "Reset All Settings" on the iPad. It erased the frozen "2" count on the badge app icon. The badge app icon disappeared. HOWEVER, the main problem still persisted because I was not receiving any new badge app icon notices. I was able to open the app and see new messages with the blue bullet point for the message box that received the new message BUT that was it - no red bubble with a # in it when app was closed.
I had a week to test out this problem which persisted 100% of the time when I received texts - whether on cellular or WiFi. Then, the 11.2.2 update came along. At first, I noticed the badge app icon appear and disappear without me doing anything. Now, it seems to be working - after testing with the second their tech, again.
To date, the 11.2.2 update MAY HAVE FIXED IT. It's been 24 hours since I updated so that's as far as my opinion could go. BUT, this issue made me want to lessen my footprint with personal tech gadgets. I refuse to waste time troubleshooting basic functionality of a device when I paid $1,000 for it.
WARNING, the second tier rep claimed she fixed it by turning on the "Text Message Forwarding" in Settings => Messages. That is incorrect!!! Even on Apple's website, under Notifications page, it explains that Notifications are independent to each devices. Also, I asked why did I have to turn on the text forwarding for the iPad if the same notifications were working with my 2010 MacBook Pro and that one was NOT turned on. Huh? What the heck? So I turned it off for iPad when we got off the support call. I suggest you read Apple's pages on Notifications and Handoff. Apple Support was no help at all.
And if Apple Support is reading this, can you please make it easier to find the list of updates for each software update? And why was this fix not listed in the software updates list for 11.2.2?