If your big tech company has an easy transfers policy + culture, you may be able to narrowly address the mismatch without affecting other aspects of your job such as comp. Perhaps your manager would even support you in making this move by helping you to spot your next opportunity and steering your work in that direction.
In any case, with the weak labor market in tech right now and what sounds like a rather short tenure with your current and first real job, I'd recommend you stick it out for another year+. There may be some things you can do to incorporate the design & coding work you love into your current role to keep up your motivation and skills, without having to overcome biases related to a resume gap or short stint when you interview for your next posting.