Is it (ab)normal to propose a course to another university as a PhD candidate
Yes, I think it is quite abnormal to suggest that a department takes up a course in their program for the purpose of research field work of a PhD student from another university. To me it seems unlikely that they are going to agree to that.

My field is not education research, but I have a hard time imagining that teaching yourself and evaluating something that goes on in a lecture on a higher level goes very well together.

Some points that you may want to consider to increase your chances of getting to do reliable field work:


Maybe it is sufficient to sit in one or two sessions of an appropriate on-going course just as an observer? Find a suitable course, explain your plans to the instructor, and I think you'll have much higher chances of getting this done than getting to teach a course yourself.
If your field work absolutely requires that you are the teacher, then I would propose that you look for a suitable course that already exists, and ask the instructor of that if you'd be allowed to teach one or two sessions in that course. Of course you have to show your qualification to do that, and it may be helpful if your supervisor could actually make or support that request (assuming that he/she is knowledgable in the field you're looking to teach in).