A biopsy can be done while breast feeding but depends on the decision of your treating doctor, the urgency your doctor feels, and also your consent. You have to discuss this with your doctor and clear out any doubts that you might be having. A lot also depends on the type of biopsy your doctor is planning, for e.g. a FNAC or needle biopsy should be the most feasible option.
The biopsy would cause some pain later on and also it could be difficult or a little more painful and tender while breast feeding. Also, the doctor might advise you to not breast feed from that breast till the lesion heals completely.
Ask your doctor the possible complications of getting a breast biopsy even though i is a relatively safe procedure.
Hope this helps.
My surgical doc was inclined to not do a biopsy while breastfeeding because of impending infection. He said it could be done, but preferred not to. I ended up having a core biopsy around 4 months later after I had stopped breastfeeding due to an unexpected pregnancy.
This is my opinion, but wait if you can. My story is one of a misdiagnosis at first. My surgeon agreed with the mammo and ultrasound that the lump I had was a lactating adenoma. Later it turned out to be cancer. However, if I had gotten the biopsy the first time, I would not have been able to conceive my little Lauren. And for that, I would not do it any other way even knowing now what it really was.
You choose, based on the information you have what is best for you and your child. Waiting may not be the optimal choice in terms of cancer, but even a small wait won't change your outcome. I might have only been a stage 2 versus stage 3, but then again...who knows.
Best of luck on the breast thing, and CONGRATS on the baby!