SIL “helping” to take care of my kids, but secretly teaching them to call her “Mummy”. Am I overreacting or is this damn wrong?

Husband and I both working full-time, corporate life in Singapore—you guys know how toxic the hours are. Recently my domestic helper went back home for good, and while we were scrambling to find a replacement, my Sister-In-Law (husband’s older sister, unmarried, mid-30s) generously offered to look after my two kids (3yo and 1yo) at our place during the day.

At first, I was super grateful. Thought wow, so blessed to have family support. I even bought her expensive bird’s nest, gave her angbaos, and covered all her transport and food.

Then last week, the red flags started showing up.

I came home slightly earlier than usual from work. The door was unlocked, so I walked in quietly. My 3yo was playing with blocks, and my SIL was feeding my 1yo. I heard my SIL explicitly telling my 3yo: “Ah Boy, say thank you Mummy [SIL’s Name] for the food! Remember, Mummy loves you more than actual Mummy who is always at work.”

I literally froze at the doorway. My heart sank straight to the floor.

I thought maybe I misheard. But over the weekend, my 3yo came up to me and asked, “Mummy, why do I have two Mummies?”

I sat him down and gently asked what he meant. He told me SIL told him that she is his “Daytime Mummy” and I am his “Nighttime Mummy”, but “Daytime Mummy” is the one who does real mummy work because she stays home with him.

WTF???

I am beyond furious, but I feel so stuck.

1. She’s helping us for free (technically, though I give token angbaos). If I confront her and she quits abruptly, our childcare situation becomes a total disaster until the new helper arrives.
2. Husband’s reaction: I told my husband, and his response was super classic SG male defensive mode: “Aiya, she just loves the kids la. Don’t think so much. She single, so maybe maternal instinct kick in. Just play along, kids grow up won’t remember one.” Excuse me???
3. Boundary violation: Is this normal? Calling her “Auntie” or even “Gugu” is fine! Why must force them to call her “Mummy”? It feels like she’s playing happy family with my children and erasing my role just because I work to provide for them.

It’s not right, but I feel so powerless right now. Every time I see her touching my kids now, my blood pressure spikes.

Anyone faced this kind of boundary-crossing in-law before? How do I shut this down firmly without causing a huge family WWII before my new helper gets her work permit approved?

Send help/advice please. Sibei stressed.