Recommendations from professors in unrelated fields
Since I plan to list this experience (is this even a good idea?) on my application, will it look bad if I don't list this professor as a reference/ask them for a letter of recommendation? 


Yes, it's a good idea. And no, it wouldn't look bad.


  If so, is there anything I can do to mitigate this? (My main reason for not asking for a recommendation is to avoid having too many letters. Is this even a problem?)


It is not a problem to have many references. Certainly having many good references can only help. 

Note: having more references than needed, while some references are not good should be avoided.