Extra work

So I have all my documents and a job offer (think ill take it) for Namyangju 1st November.
I understand that all private lessons are illegal but that it is possible to teach at other institutes as well if you pay for a visa extension thing and get permission from your employer.
I noticed on craigslist that there is a part time, temporary  position sort of Improving the fluent English for a group of employees who are about to go on a buisness trip to England.
The teacher would be employed by the company.

Im probably not gonna be applying for this position but i am qualified for it and it would not interfere with my hours at the hagwon. I wouls like to ask a few questions

1- If i got the proper authorisation from the immigration department would this job be legal for me to do (would it count as private tuition?)

2- I understand that my Employer would be unhappy if I asked to work for a potential rival but would she be upset if I asked for permission for this job (i.e. teaching a class of people who could not otherwise enrol at her facility).

Even if jobs such as this are very rare I would just like to understand my options.