A first indicator of what this app does is it's package name: com.google.android.ims
IMS is usually the abbreviation of IP Multimedia Subsystem, which is dedicated to do communicate with the carrier over IP, instead of special protocols as this was in 2G/3G.
Another hint gives us the description in Play Store:
Carrier Services enables the latest communication services from mobile
carriers, including battery-optimizations and support for enhanced
features in the Android Messages app.
Therefore this package seems to implement the client side of "RCS Messaging", the successor of SMS.
I am not sure how many carriers already support RCS and how widely this is used. In general I would allow background usage, otherwise the app would be pretty useless if you every receive an RCS from someone (you never know who might send you something).
From my understanding plain SMS should not be processed by this app.