A useful test to help understand the condition is a thyroid scan. However, the test requires that radioactive iodine be given to the mother and this material must not be given to nursing mothers. The radioactive iodine will be found in the milk for weeks, and concentrated in the baby's thyroid. There are ways of dealing with postpartum thyroiditis without doing this test. Although surgery on a lactating breast is more difficult, the surgery does not necessarily become easier if the mother stops breastfeeding, as milk continues to be formed for weeks after stopping breastfeeding.
Mammograms are more difficult to read if the mother is breastfeeding, but can still be useful.
Evaluation of a lump can be done by other means besides mammography. Discuss options with your doctor. A needle biopsy, for example, can be done of a lump which is of concern. There is no reason that you cannot continue breastfeeding if you become pregnant. There is no evidence that this does any harm to you, to the baby in your womb or to the one who is nursing.