Can we extend this contract in this manner? Would we be better off to issue a new order with a justification under FAR 6.3?
You may have more than one option. Assuming the company wants to continue performance, you might have time to make a bilateral modification adding the extension clause. That will give you 6 months to solicit and award a new contract.
Since its 2 weeks until the end of the contract, it is more technically sound to create a new order, preferably one with 52.217-8 included. Again, if you invoke Far 6.301-1 or 6.302-2 for justification of a short solicitation period, you may need to consult your legal department. Legal may also ask some questions as you seek a sole source justification. Why was FAR 52.217-8 omitted? Was it left out on purpose or by mistake? Was the contract meant to end after the fifth year? In any case, a new action with the clause should solve the problem.