I agree with your approach, and the loading / axle configuration you have proposed sounds reasonable. I have worked on several smaller private (and sometimes public) bridges where the AASHTO loading provisions are not applicable and engineering judgement must be used as to how the bridge will be loaded, what restrictions to loading there will be (physical barriers/geometry vs. posted signage, etc). As BridgeSmith states, it is up to the owner to stipulate what design vehicles and loads they would like to use, but typically most private owners of this nature won't be able to tell you more than something generic such as "dump trucks and concrete trucks," or "legal loads" and rarely will you get any defined axle loadings from them - they are looking to you as the structural engineer to provide this guidance.
Use engineering judgement and owner's input to determine the design loading, check a handful of load combinations that may control, develop an envelope for the loads to be resisted, and design accordingly. It sounds like this is your approach. The process is simplified from your typical DOT bridge design and lends itself to hand calculations, engineering principles, and sound judgement.
To BridgeSmith's second point, if seismic is a big consideration, the need for a more rigorous analysis and consultation with a bridge engineer is advised.
Final thought - load posting would be a good idea as mentioned by others. And often times we will have a "one vehicle at a time" sign coupled with a 5 mph speed limit sign to minimize impact loading (I would still design to account for impact - generally limited to ~30% increase to the axle loads).
