It's not about competition with the MIL though...
OP suggested husband sees the MIL too that day and clearly gets on with her.
I'd be more worried about husband seeing himself as "seperate and single" and picking fights for no reason, and redefining his family unit as "something he was grudgingly forced into" even though he's been enthusiastic before.
As a single woman, I find married fathers can "try" to start sighing at me
They want to tell me how hard it is looking after their own child, and it's all the needy harridan wife who actually wanted the family and they're "her" babies and she's forcing him to do couple stuff theres always a fucking sigh
(I run like hell, and assume that there's an attractive wife doing 24/7 who is perfectly pleasant and easy-going).
I'm sure and hope it's just a blip and a lovely lunch occurs whilst we're all talking nonsense on MN.
But this sort of thing does happen.
OP how independent are you financially? What's the childcare payment plan? Is husband paying for his child? Is he making looking after the baby "your job"?
Better safe than sorry, hope for the best, plan for the worst.