The owner of the copyright to the source code is the one who must make the decision on how to license the code, if any license is going to be provided.
You have a number of misconceptions behind your question.
As a publicly funded body, I am under the impression that anything that doesn't conflict with the Data Protection Act 1998 should be made available under the Freedom of Information Act 2000.)
First off, I think it's a stretch to consider source code as data. Source code is compiled into instructions that operate upon data. The source code executes and enforces the school's operating procedures.
Second, there are a number of exceptions within the FIA that you reference. You already cited one potential restriction within the code that would prevent disclosure, but there may be many more.
Third, the FIA is based upon citizens requesting information. I think it's a stretch to say that the FIA prompts you to preemptively release that information.
Fourth, FIA enables transparency so that the citizenry may review the actions of the government and other public bodies. The FIA does not provide a license for the reviewer to then re-use that information in any way that they see fit.
I believe it should be open source -- and feel that the MIT license would be most appropriate
MIT license essentially gives away the source code for free and allows the recipient to do whatever they wish with it. Even if the source code is subject to the DPA and FIA, it's a really big stretch to say that the code should just be given away freely for anyone to do whatever with.
This is largely moot, because the source won't be published publicly.
Then why are you licensing the software? A software license allows the copyright owner to grant permissions (a license) to a user of the software (licensee). If the software won't be published, then others can't access the software and would not be able to utilize the permissions (license) that had been granted.
This is probably something that should go to the legal department
Yes, as a matter of fact it should. Because the legal department represents the legal interests of the University, which is the copyright holder. As an employee of the University, you are not the copyright owner of that source. Therefore this is not your decision to make. It's their decision to make, which is the function of their legal department.
presuming they have any inklings about software licensing
Is a moot point. Courts have consistently upheld that a property (copyright) owner does not need to have a full understanding of all of the ramifications in order to make decisions about their property. The fact that you don't believe they understand what's involved is irrelevant to who should make the decision and whether their decision is binding.
but what software license should I use against code developed for use at a publicly funded institution
Unless the legal department tells you otherwise, you shouldn't be putting a license on the code. You could put a copyright statement in the code indicating that all rights are reserved and the copyright belongs to the University. However, the Berne convention considers the equivalent of that to be the default copyright statement and states that you are not required to put a copyright statement in place. You may, and it's probably better if you do in order to make things clear, but you're not obligated to.
With all that having been said, kudos to you for thinking through this issue and asking before you published the code. In theory, if you had publicly published the code then you could have been reprimanded for releasing University property.
You do have some valid questions that the legal department should answer for you, but please realize that Intellectual Property can be a fairly complex legal field with a huge number of exceptions based upon the jurisdiction.
The best thing for you to do at this point is to contact your legal department and ask them how they want the copyright for the code to be indicated. Also ask if they want a license applied to the code and bear in mind that the code need not be licensed at all.