The administrative boundary in OpenStreetMap is represented as a blue and dashed white line. More can be found here: Tag:boundary=administrative - OpenStreetMap Wiki. (Administrative boundaries for local government areas are typically categorised as admin_level=6)

Add a new node onto the road you want to split to the LGA boundary. Then split it (right-click on the node split)

Select your newly created node (white), right-click and split using the scissors icon

Select the node to move it

Move the split (grey) node onto the blue boundary

Select the road segment (or segments) where you know the name of the road operator. In this example, it will be Yass Valley Council. The tags to add onto the road segment for a local government road manager will be operator=Yass Valley Council operator:type=government. Note: you will need to know the name of the operator in your circumstance, i.e. operator=name of road operator

If needed and known, you can select the other segment of the other side of the LGA boundary and repeat the previous steps, filling in the appropriate tags for the road operator name and type. You can save and add a changeset comment such as “Split road to reflect change of road operator at local government boundary”