If you can import your list into the qualtrics loop and merge function you can use display logic from a question recording a participant's identified gender (e.g., Answer to question 1 is male then show show Block A; Answer to question 1 is female then show block B; Answer to question 1 is other/non-binary, then show Block C). Then in Block A, B, and C you would use the loop and merge function to import the gender-appropriate names. You can select the 'random' function in loop and merge and indicate that you only want it to show 1 of the potential names. You would do this for each block with the right names.
the downside of doing it this way is that when you export your data, you will have a column for every single name a participant could have seen, even in the other blocks. However, they will all be empty or -99, depending on how you treat missing data. For the name the participant actually say, there will be real data there (e.g., text they input or selection they chose).