I did Intern/Co-Op. Full time during summer and part time during the semester. Department dean don't know much , you need to convince him/her. Take that CPT form to the Dean and mention that, its a Curriculum Practical Training where international students can work and study together if the work area is related with the major.
Its a full time since its summer, so its not a big deal.
Or what you can do is :
Tell the employer to write another offer letter mentioning that specific word " Intern" position. Mention that, you will be working full time, but this stupid dean is not letting you sign the form. And, you will be all set to go.

But, remember, Dont let intl student know that , you are taking classes in the summer and working full time. I don't think so , you can work full time when you have classes in the summer. But, I guess, 2 classes should be fine.

However, since you are taking only 2 classes in the fall, you can still work full time since its a final semester of graduation.

Tell the employer to write another offer letter mentioning that specific word "intern" although you will be full time through out the year. They should be ok to do that.

Good Luck.