gentvenus

Mentioning the contact you had with the professor will certainly help. If he likes you enough, I'm sure he can pull some strings to get you in there

IMO it would be better if your let the professor know beforehand so that they know what you are trying to achieve.
Hoping your plans will come up "naturally" during the conversation will often result in unnatural behaviour on your part as you will always be thinking (at the back of your mind) whether now would be a good time to introduce your plans. The anxiety would almost be unbearable!

It causes me heart palpitations actually ?

I see. Thanks for sharing! Clearly you are not at fault for their own administrative timing delays. They should waive the overborred amounts! ?

I had the impression that most schools have such support schemes. Whether they are sufficient, however, is another question.

Very inspiring motivations there! As one working in public sector myself, I can safely say it is your passion for helping others that will get you far. Anyone joining the public sector for any other reasons will find the journey very tough and unrewarding.