This sounds a lot like a layoff, not a termination for cause. Just keep the letter so you can prove it.

Now you still have the issue of moonlighting. Did your company policy allow that, or did you get permission? A problem may arise because PRIA (federal law) requires that you report all aviation employers and the flight school will provide your dates of employment as part of that. The issue isn't that a part-time CFI got laid off, it's that you have a paper trail showing you were moonlighting (and possibly violating flight time limits?)