Would this make a good case to hire a lawyer on?
You have a viable claim of breach of contract unless the [bonus] offer made to you is bizarre enough to exclude hires which exceed three months. But the focus on whether or not to retain a lawyer is pointless because you might end up hiring one who is too expensive, incompetent, or both.
The first thing you need to ascertain is whether the amount owed to you shall be litigated in Small Claims court (if that forum exists in your country or jurisdiction). Each jurisdiction specifies the maximum amount that shall be litigated therein.
If your matter has to be litigated in Small Claims court, then you need to figure out whether parties are allowed to be represented by a lawyer. Some or many jurisdictions require the parties to represent themselves, not through attorneys.
If you are allowed to retain a lawyer (be it because your matter is not within the jurisdiction of Small Claims or because that forum permits lawyers), then you need to ponder whether your net compensation --that is, after deducting attorney fees-- is worth the likely deterioration in your relation with the employer (for instance, what if the lawyer wants to bill you $5,900 for his services?).
If the doctrine of at-will employment is applicable to your context, you also need to assess the likelihood that the aforementioned deterioration will prompt the employer to terminate you.
That being said, I would never discourage from pursuing your viable claim. The fact that your employer is defrauding you on this issue shows his propensity to defraud you again in the future. I am only pointing out some of the risks you need to weigh so that you make an informed decision on how to proceed.