Keep going OP, if your MIL keeps on in this way just tell her where to go - elsewhere - if she doesn't want to see her grandchild having a bit of nosh. Or, if needed, tell her the ruder version of where she can go.
Personally, I have bf 4 boys. They fed anywhere and everywhere, but I sometimes used a shawl or muslin to cover my flabby bits if I wanted to, but that was normally any belly showing when having to pull my top up as I was self conscious about it. But I also seemed to be naturally discreet without trying to be, so many people didn't realise I was feeding anyway.
When ds2 was days old I was at a little toddler group at the local toy library with ds1 and fed ds2, a grandma asked for a cuddle of ds2 and I had to disappoint her by telling her he was feeding. She hadn't realised as I was pretty adept at quickly getting sorted by then so no one had realised he was feeding. She did eventually get a cuddle.
Thankfully my mum, her husband and my MIL were good with the bfing. FIL made a couple of comments about it early on, but that was his way of dealing with his embarrassment of possibly catching a glimpse of his DILs boobs. I think once he realised I wasn't wandering round with them out in his face all the time he settled down. It probably didn't help that when his twins were born in the seventies it was assumed they would be formula fed, even though MIL wanted to bf, but she was given tablets to prevent it then they FF, so he hadn't witnessed his wife bfing. By the time grandson no 4 came along he was much more relaxed about it.
I did most school drop offs and pick ups and bfed on demand. Sometimes ds2, ds3 or ds4 couldn't wait the 45 minutes until we got home so they got fed while walking to or from school, or even on the playground. They fed in other people's houses, in cafes, restaurants, McDs, in the odd shop, wherever we were if we wouldn't be home home in time to avoid a screaming tantrum because of hunger. Oh, and we've been churchgoers for many years, that is where we met, and often fed babies at church. Any church that doesn't accept bfing is a church I wouldn't want to be part of tbh.
Don't let your MIL ruin what should be a special time between you and your baby, if she carries on with her stupidity tell her she is welcome to not be around you when you're feeding your baby. It's her who will miss out on spending time with her grandchild then just because of her weird views.