I missed arbitration for the first time and now I have my best pitcher giving me the "After taking me to arbitration 3 times, I'd prefer to play for a franchise that treats its players better. Free-agency, here I come!" Now I can't even submit a contract offer to him.

Is there anything I can do to submit an offer during the season, or am I just going to have to compete with other teams during free agency?

You, personally, can't offer him anything, and he won't sign with you during the exclusive re-signing day next season, either. However-- if you trade him to another team it is substantially more likely than not that he will re-sign with that new team. So you can potentially recoup something near his full value in a trade.