There are two avenues which could be concurrent or separate that Bauer could pursue.
To make it very simple at a 30,000 ft level- The Dodgers stopped paying him and they should have continued paying based on the contract.
That avenue will be a lot more difficult to pursue due to the contractual language with MLB about violation of rules and the MLP agreement etc.
The second avenue, sue MLB for slander (reputation) and lost revenue due to the slander. Basically, MLB found him guilty regardless of lack of criminal charges and having evidence showing exonerating him videos and texts etc. MLB slandered his reputation vis a vi the suspension thereby causing MLB teams to not want to offer a contract and loss of revenue/ability to provide via his profession.
Bauer will be able to show that pitchers with similar or less metrics were getting X $$s and considering he was one season removed from a Cy Young he should have been a hot commodity.
The open question is what are the damages, you need to know that before you proceed in any case. You need to show standing and state how you were harmed. How Bauer was harmed, is yet to be determined... If he signs a contract for an SP of his value in the next month for an MLB team, Bauer could only claim lost wages between when the Dodgers stopped paying him, minus the value of what his MLB market would have been and what the Japanese team paid him.
I assume an arbitrator or the courts would use MLB arbitration numbers to see what a pitcher like Bauer was worth last year.
So to summarize, way too early for Bauer to pull any triggers or go nuclear..