I don't think Dr Iwai -could- do anything even if she wanted to [and I believe she would because she is very nice]. It's a very high demand class. Keep trying on Solar as well [I have no idea how waitlists work, but if you're on one and you're trying on Solar too, higher chances, ne?] it might open up. Even after classes begin keep trying until the deadline. Especially the first/second day of class.
Also, it's offered in the summer [and is easier to get into, if that is an option for you]. You don't choose exchanges until 3rd year, so technically you could take first year Japanese in the summer and second year Japanese in your second year, making you no less behind than anyone getting in during this semester
EDIT: I took your profile saying you're in your first year as meaning you're just starting this semester. But as you've taken Mandarin last year, it stands to reason you're in your second year making my "you'd not be behind with summer school" sentiment false.
You could do an exchange in fourth year? [Is this a possibility?]