I agree with Dr Ho that you should seek a doctor's opinion early regarding your eczema, especially if the rash is itchy with signs of weeping.
It sounds like you may be having an acute flare of your eczema.
While the eczema itself should not affect your breast feeding in terms of the quality or safety of the breast milk to your baby, breastfeeding especially with multiple feeds a day regardless of direct latching or pumping may cause the skin to be more dry and sensitive, especially around the nipple area.
This may in turn worsen the underlying eczema that you are already experiencing.
Eczema is best treated with regular use of moisturizers as well as topical steroids, which are generally safe.
However, it would still be good for you to seek a doctor's opinion first to ensure you receive timely and appropriate treatment.
Hope this helps!
