Electrical – Service box with no neutral bar

Yes, you can add an insulated ground bar in the can (you should be able to get one specifically designed for your particular can, but here's what they look like):

.. and then connect it to the other ground bar using a piece of #4 AWG (solid bare copper, or stranded THHN insulated) assuming you're still at 200 amps. This guarantees that the green bonding screw that's actually bonding your neutral on the other bar will affect both bars if it is removed. It's common to add these for isolated grounds, where they're connected to a different ground altogether instead of the other bar.

Note - the one you buy may come with its own bonding screw, and the appropriate tapped hole to receive it - so check for that prior to connecting the two together (IOW, read the installation notes that came with it).

The neutral and ground MUST NOT be bonded at a sub-panel. They should only be bonded at the main service panel. If you bond them anywhere other than the main service, the neutral return current now has multiple paths, including though your ground wire.

You should be able to buy a second bar for the sub-panel if it really is meant to be used as a sub-panel. The neutral bar will need to be isolated (it should have plastic insulator separating it from the case). The ground bar should be bonded to the case.

The ground wire from the house must be connected to the sub-panel ground. Since it is a detached building the ground rods are also required by code and should be tied to the sub-panel ground. The ground to the service is the most important because it will provide the low resistance path back to the service in case of a fault.