The tuberculin Mantoux skin test contains phenol or carbolic acid. Some people are allergic to this substance.
Depending on which country in the world you were born, if you received the routine BCG vaccine at birth, you WILL test positive with this test. It will most probably be a false positive which will require you to take another test.
You can ask to take a chest Xray, a sputum test, or a Quadriferon Gold blood test for the detection of tuberculosis. You may have to pay out of pocket for the test, but these tests will be safer than the skin scratch test.
Let us know what happens. I have done the sputum test when I am paying for the service, private practice. If you are in a prepaid plan, they may TELL you what test they will do as their protocol calls for. I was at the VA and asked for a current TB sputum test while I was there as I was due for one for my work, and they told me that they had a protocol to follow, and I could not just ask for any test. I offered to pay out of pocket for it and the doctor did no know what I was talking about.
I personally would not be concerned about having a Mantoux test whilst breastfeeding. However, if you've had a BCG in the past, the preferred test is a blood test because, as applejuice said, you can get a false positive Mantoux following BCG vaccination.