I found a solution. Apparently it had to do with Private Relay / Limit IP address tracking. I’ve turned it off and now the app works again.
See here: Iphone (IOS 16.6.1) ignoring DNS server s… - Apple Community
And here: Manage iCloud Private Relay for specific websites, networks, or system settings - Apple Support
I’m still confused why accessing via the browser on my iphone kept working, as well as the HA app on iPad where Private Relay is also turned on 
[Edit] Now that I know the issue, also found this older topic covering the same issue and solution: After updating, iPhone app will never use internal URL - #11 by activescott