For purpose of this particular scenario and again
functionally applied...
Cross-deputization doesn't change anything and does not expand jurisdiction - only options. It allows the officer to enforce rules of both legal constructs. A tribal officer that is cross-deputized as a state/county/city official can apply tribal law to those of it's jurisdiction, or state/county/city law to those of it's jurisdiction. If a non-native commits a tribal CCW violation they are still not subject to tribal jurisdiction, only state/county/city applies. If no state/county/city violation has occurred, it's not actionable.
Think of it in terms of traffic: A state highway bisects a reservation and there is parallel law in both tribal and state code such as a speed limit. Tribal officers can cite natives to tribal court, and then non-natives to state court. This isn't uncommon. Similarly, state/county/city officials can also be cross-credentialed as tribal officers with the same effect, but that is less common usually for reasons of sovereignty and local politics.
As @
Chuck Whitlock notes, it gets more complicated with some types and levels of violation - such as VAWA, or civil matters. There's also some quirks in intersections of federal, state, and tribal fish and wildlife law.
There's an enormous amount of bad info and instruction out there on this topic.