Hello, I hold both an AT and PT license (2015 and 2018) graduate. If you want to work in sports having your ATC and DPT looks 100x better than holding a DPT and the "SCS" specialization, so I highly recommended the athletic training completion if you wish to work in professional or D1 level college sports. However, things are different now as the athletic training programs are all switching over to masters level instead of bachelor level (what I have). This means you will be earning a masters in athletic training which again helps you look more juicy to any professional or D1 sports team.
So to answer your question, can you work while completing ATC school- Yes. I worked as an athletic trainer on weeknights and weekends during PT school, including during times where I was a full time clinical student (I was very busy). However, I do not think a full time job will be realistic, only part time. Just as in PT school you will have lecture, lab, and clinical rotations, meaning you wont be able to work 40 hours a week.
My suggestion would be to take a PRN hospital acute care job (I know its not sports related but at least its good money coming in) and it allows you to work weekends and holidays. Another option which depends on where you are placed for clinical rotation would be working part time in an outpatient (hey sports woohoo) clinic in the morning for 5-6 hours, then after lunch heading to the high schools to cover your clinical rotations.
Sounds like a lot or work ahead of you but at least after completing DPT school AT school should be a little bit easier
