Can I just flat out ask my coach for a coxswain seat race? How do I go about asking such a question?
If you wanted to, I don’t see why not. Before directly asking for one I’d first talk with your coach to find out why you and the other coxswain(s) are in the boats you’re in. Assuming this is a “I’m in the 2V but wanna get in the 1V” type of situation, I’d get their thoughts on what they think makes you a good fit for this boat and/or what’s keeping you out of the other boat – i.e. what areas could/should you improve on that would make you more competitive and give you a better shot of getting in there.
The key is to talk yourself up instead of talking the other coxswain down. Explain why you are the best person for this boat and why your coach should consider giving you a shot at coxing it. If you’ve been working on a particular skill and feel like you’ve made some good improvements, tell them. If you think you can get in the boat and make it faster because of XYZ, tell them … but also show them by doing those things with the boat you’re currently in.