The way I see it - the "IT Department" is really just you because the programmers are more akin to an "Engineering" or Tooling department IMO. Are the programmers providing IT support? If so, this is a double edged sword for obvious reasons.
I have worked in offices only slightly smaller than that company and we needed at least two people most days - and we had the benefit of having outside help for a lot of things (having a high staff turn-over didn't help).
I think it's worth making a business case focused argument rather than a "we need help" based one. Perhaps you should get the help of a manager who is not in the "IT Department" to help build, mentor and deliver the case. This isn't necessarily because your existing IT Manager is incompetent, but mostly because he is too close to the issue at hand and is unlikely to be taken seriously because of it. He also sounds like a typical tech guy - and thus probably isn't quite as tuned into non-IT culture.