Chi Square: Columns are repeated measures of ordinal dependent variable?

#1
I have 300 people. They are placed into one of two groups (Group_Variable: A or B). Then four measurements are taken (V1, V2, V3, V4) and these are on an ordinal scale (1-3). [The four measurements are on different parts of the body but should theoretically have the same value - for example, skin darkness]. I want to assess if the measurement (V) is different between Group A and Group B. To get a feel for the data I created a chi-square table with 1200 data points. But I feel like this may not be the most appropriate test as each person is contributing four data points, and I thought some type of repeated measures analysis would be more appropriate. However, all of my searching seems to lead to McNemar or Cochran, neither of which seems applicable to my scenario (or I am misunderstanding how to apply them to my study).

Thanks in advance,
Mark