I'm a student and I have occupied a position shortly (work less than 7 days in total over 4 months) in a franchise during my studies a few years ago, but didn't make through the probation period, which I felt was due to the lack of trainings and my lack of motivation (I didn't received any training there and was too busy with my studies to bother telling the supervisor). Since then, I had a few more short-term summer job experiences elsewhere, but with better references and experiences. I also had similar short term experiences (a few days of work over a period of a month), but decided to keep it on my CV since I made a good impression, but the reason for my termination was out of my control (they were overstaffed)
I've got a new offer for a much higher up but related position recently in the same franchise, but I omit to put my first job on the CV since I am now much more capable than before and having it on my CV would make me seem like I am unable to hold a job.
They have already contacted my references and gave me all the paperworks. I'm concerned however that my omission might show up in their third party background check once I have submitted said paperworks. They have not asked me for an employment history, but they did asked me to sign a background check form which says that any omission on my CV might be a reason for termination.
Should I tell my employer prior to the background check in good faith? If yes, how should I approach the situation? If no, what are the likelihood that they will find out and how will it reflect on me as a employee?
thank you very much in advance