Some people believe in making guests comfortable by bending to suit them, some believe guests should abide by the house rules. Most are inbetween.
I don't think this is about following rules or welcoming guests.
It's about the baby's health and the mother's well being. The baby is lactose intolerant and that means it is important that the mother's breast feeding relationship is supported and protected. 50% of lactose intolerant babies will also be intolerant to soy formulas.
If the MIL is intent on making things difficult, upsetting and alienating the DIL then she is and it won't matter what the DIL does. However, I don't think I could brazen it out if I were her and I'd get out of the environment ASAP and consider limiting contact with the MIL for the foreseeable, as someone who puts irrational offence taking over the health of their DIL and a baby who is already struggling is not someone I'd want around my DC.
But I can't emphasise enough to those who are saying it isn't a big deal and she can go up to the bedroom or that she is looking to take offence or that people should tolerate offence taking that how reasonable any of those lines is depends on the other factors involved. The other factors involved mean that this MIL is really out of line IMO.
Because of a newborn's rapid growth and brain development being underweight is extremely worrying and it is really important that real care is taken with feeding and that it is much better, health wise, if those vulnerable babies can be breastfed. Most people give up because of lack of support. That's why this is so important. Feeding a vulnerable baby is not a normal experience of breast feeding and one that you need more support with.
Breast feeding in public laws are really important because the sniffy attitude of folks like this woman is responsible for a lot of women struggling or not trying in the first place. I don't really care if sometimes people have archaic attitudes or don't know why they are offended.
When it is something as important as infant feeding, and especially feeding of a prem or underweight or ill baby, I think the presumption is feeding goes anywhere even in other people's houses and the offence takers have to learn not to take offence. That is a significantly better thing that breast feeders not feeling supported because lack of breast feeding has a cost to society and to health across a population and not being offended by it has no negative consequences at all and in fact some positive ones for society.