The only real reason for doing an mtDNA test for genealogical purposes is to prove that one person's female line connects with another person's female line. However, mtDNA is passed down almost identically from generation to generation with individual mutations happening rarely. So you will be unable to tell how far back the two lines connect.
If you know your common ancestor with your 1c1r is on both your maternal lines, then there is no need to get your cousin to do a mtDNA test because it will be identical or maybe have just one mutation. If you don't know, or suspect there might have been an NPE (Not the Parent Expected), you could choose to get your cousin to do the test, as that would prove or disprove the maternal line relationship.
Autosomal DNA will definitely tell you that a person is related when they are as close as a first cousin, once removed. If you have any other close relatives tested, you will be able to tell through triangulation whether they are related to both of you and then you will be able to impute who the common ancestor is.
So your first choice and best bet if you have other relatives that have tested should be to get your cousin to do an autosomal test, and also try to get other relatives you know are on your mother's mother's side tested. For a relationship that close, you should be able to figure the exact connection.
If from that, you find out the relationship, and if you find you are both connected on both your maternal sides, then an mtDNA test of your cousin won't provide any added benefit.