Should I tell my potential employer that my graduation date got pushed back to August for 1 outstanding gym credit or keep it to myself?

I'm on my last semester and received a job offer to start full-time employment at this firm I've been interviewing for, about six times since January. I will successfully complete my degree requirements for my major, but I will have to take one summer gym course that last 6 weeks to finish my degree. The timeline of everything is as follows: I graduate in May, start the gym class that occurs on Saturday and Sunday for 6 weeks. and by the end of June, will have earned the credit to get my degree. The problem is, my degree is going to take 4-6 weeks to get to me after graduation on May 19. This is problematic because it will be pushed back to September graduation, and be mailed out then. The dean has offered to write a letter to give to my company to tell them I have completed my degree requirements for my major, and the gen ed requirement is outstanding because I have one credit left for it.

The gym credit is outstanding because about two weeks ago(this semester I have been taking my gym course to finish that credit) I get an email about how I've missed 4 gym classes and per syllabus I can only miss two before being unable to pass the class. I emailed the instructor about my interviews for THIS job and she never once complained or told me anything other than good luck and thanks for the heads up. So the dean complementarily dropped me from the course and offered her support to write the letter because she can't change the athletic department's policies. It's bullshit they are so unwilling to work with me. My question is, hopefully for hiring managers, managers, and directors who employ or have selected employees, do I tell my hiring manager or future boss about my degree being pushed back? Do I keep it to myself until I get it since by the time I scan a copy of it and send it to HR it would be an authentic letter of degree completion? (<5 weeks after my start date). I really don't want to lie to my future employer, he has been so kind and upfront with me about the job and starting off on a lie will cause me a lot of guilt and shame. However, it would be horrible if I get declined the offer because of this as I have signed an apartment lease and am preparing to move at the end of May to be closer to the job. I'm scared and I don't want to keep the information to myself, but I also don't want to be a liar and get fired if they find out or need my degree before the first three months of employment. There could also be jail time or something if they pressed charges right?

I am in Ohio and want to know from previous hiring managers, bosses, directors, and other people who may have more understanding of the hiring process and degree importance, as well as if there would be legal consequences for not sharing the information with my employer. Will the background check reveal this? Will they be willing to hire me still if I tell them? Would you fire me or keep me? Please give me advice, I am a first-generation college student and I finally feel like my work has paid off and bam, it's like its all being taken from under me for ridiculous reasons.

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