This looks like it's all very messy.
There are a number of things you have to consider. You do have to ensure your concerns are recorded at the correct level, at work. These interchanges between yourself and the new head chef may be “forgotten” by at least one of you, if things unpleasant. In many circumstances, where a grievance is brought, the approach can be, “if it isn't written down, it didn't happen”. Harsh, but so very easy to achieve on your part. So, you do need to raise your concerns officially, and you really need to do this in writing. It may not be a legal requirement, but it is the professional approach and could save frustration at a point in the future.
But, in the meantime, I would suggest you try to think about how you can minimise the impact of your breaks on the workings of the kitchen. I don't know what you actually do, or what any impact might be of you not doing those things for a period. For instance, if you cook eggs at breakfast time, who would cook them if you're having a break? Probably a bad example, but if the answer is customers wouldn't get eggs, then that's going to get tricky. It would not be considered reasonable to expect your employer to suspend the sale of egg based dishes while you have something to eat yourself. Now, that's probably a very bad example, but hopefully you will see what I mean. So, a reasonable approach could be that you can only have breaks within certain time slots, which might not be your preferred times, but your employer is there primarily to make a profit. Impinging on his ability to make a profit is not going to end well.
Thinking longer term, I think you have to consider the impact of this whole process on you and the head chef, who could have some changes enforced upon him, quite publicly. Will that relationship settle into a healthy and mutually respectful working relationship? I have no idea. I am absolutely not suggesting you walk away from your job. As you so rightly say that would have a number of knock-on impacts. However, I would suggest you seriously look for something else where these issues would not be such a hurdle. That way you are in control.
Good luck with it all, and let us know how you get on.