No a dedicated, non-player, host does not need to have individual aircraft modules installed to host missions for said module.
As long as the Tarawa is "freely" made available to the core DCS engine then a dedicated host would not need that either. If the Tarawa requires ownership of the Harrier module or is a paid-for DLC then it would need to be "owned" by the dedicated host account.
I believe based on what I last remember reading was that the Razbam ship modules would be included with the Harrier module. What exactly that means for your dedicated host, I'm not sure.