Thanks for the reply, but this isn’t what I meant. Maybe I didn’t explain it too good. The problem I was having is for a custom field, not a front-end form.
I figured out a solution on my own. I made added a save_post function, like you would if you made a custom meta box to save the hidden fields. I just called that action from the constructor.
It seems like maybe one of the built in functions from the custom field template would have been better to use, but I could figure out which one, if any I should be using.