You don't own the source code for the project, so you shouldn't be the one making this decision.
The fact that you were the last one to contribute some code doesn't mean that you have the right to slap a license on the whole thing and dictate its further use. In fact, even if you wrote the whole code, copyright is almost certainly owned by the university. The university legal department is probably who actually owns this decision.
Authority to decide the licensing might be officially devolved to the computer science department. Or it might be that CS is just doing their own thing because nobody really knows what is going on. You also might get away with just putting a license on it yourself. Someone in the department might even tell you that is fine because they don't care.
The problem with such decisions taken without the real authority to take them is that someone might start caring later--for example, if the software becomes a successful product and somebody (not the university) is making money on it. Many open source licenses (including the MIT license) are incompatible with the university retaining rights to commercial use. The university would have to explicitly decide they are ok with giving up this right for such a license to be ok.
Your own self-interest in the matter complicates things.
If I understand you correctly, you want to open-source the project at university A solely so that you can take it to university B and use it (and you don't want to share it further after that). Frankly, this doesn't look great. If there is any ambiguity in the licensing, or any dispute arises, the fact that you did this for your own benefit rather than sharing with the research community is likely to color the whole thing. Any benefit of the doubt will be gone.
I would not do this. Instead, explicitly discuss with the project your desire to contribute (and collaborate).
Rather than open sourcing as a means to an end, get what you really need: permission to work on the code and collaborate with them from another institution.
- They might decide that the best route to doing this is open source--in which case let the department lead that.
- They might decide that simply giving you permission is the right route.
- Or they might even say no--in which case be glad that you are finding out that answer this way rather than in a much more contentious way later.