J.G.
Medically/scientifically speaking.. I don't know.. but I did notice I was not able to produce as much or the let down took longer when I had my periods when nursing. I think the kids did get fussier then because they weren't getting as much. I had to almost meditate and find a way to really relax during these times and it helped some. Try drinking even more water during this time.
With my #4 child I had a difficult time producing milk from get-go, it was suggested I take fenugreek and another herb.. I cannot remember the name - I want to say thistle something but not the milk thistle. I am not sure either of these can be taking while menstruating but it couldn't hurt to look into it.
you might be surprised at what foods can bother a kids.. and as they get older their taste buds could change.. and whatever you eat can flavor the milk so your baby might not like the taste.. or digestive-wise. their gut might not be able to take it.
for example: I noticed when I had garlic and onions my kids got really fussy and wouldn't nurse.. they didn't like the taste.. and they got really gassy. I did find that I could eat cooked onions though. so it might be worth keeping track of what you are eating and seeing if you notice a correlation with certain foods. As a general rule of thumb - if it makes you gassy it will make the baby gassy.. but not always the case each kid is different. Strong flavored food will flavor the milk to. Chocolate is supposed to make them gassy too.
So consider foods... and the possibility of lower production during that time of the momth.