Second part time job in different profession in addition to ongoing Tier 2 ICT

[VolkanD]

Hi,

I am a valid long term Tier 2 ICT holder since 2015, continuing to work under my sponsor company.

I recently received an offer for a part time tutoring/supporting instructor role at a university, that will require my involvement several hours per week. I am very interested in taking this position in addition to continuing my current job with my visa sponsor without affecting my current job (and my sponsor is happy with this arrangement).
However the university job is an academic teaching position and my current job description in my CoS does not cover this explicitly (even if I would be teaching subjects based on my professional experience in my current job with my sponsor company). Because of this I suspect that the university contract would not fall under the "supplementary employment" category. Is this a correct interpretation?

If yes, can I apply for a separate visa or permission to do this secondary job, while somehow keeping my ICT/current job? I understand that at this point I cannot switch from ICT to another Tier 2 visa without the cooling off period outside of the UK. The part time job on its own does not meet the requirements for Tier 2, either. Therefore it appears that I am out of options, however I thought I would seek expert advice before refusing the offer that I would otherwise like to take.

Thanks in advance.