Op, looking through this from a cultural perspective it is clear that your DH never ever has any intention of asking his mother to work and you will alienate yourself from him for even suggesting such a thing.
So I suspect an uncomfortable likely truth in the matter which you won't be happy to hear. But please hear me out.
I think it's highly likely your DH has known all along that he will never ask his mother to work.
Your DH knew he would need to marry wisely to help share this burden.
So he married you, not just for love but also for cultural expectations.
You are used to working, indeed you expect to work and you are happy to be the modern working woman helping to provide for your family.
You have a financially sound background, you are well qualified and you are earning a good amount of money.
This would have been an attractive proposition to both your DH and his father and the rest of the family on his side.
They would have discussed his marriage to you as financially indirectly supporting your mil by default which would relieve fil of this pressure.
I believe your fil and your DH are in cahoots about this.
Your DH playing ignorant about his mother's lack of pension is a white lie ( in his eyes).
He wants to keep you on his side so he will say whatever he can to keep you on his side.
But this pursuit of his mil to work will be preposterous to him and he will not contemplate it and never will even if he has gone along with the idea in theory to deflect the attention away from his role in the status quo.
He won't be honest with you on this matter and he is unlikely to admit, oh yes, your earning potential was the deciding factor for me to marry you in the first place.
To be fair to him, it might be a surprise to him that knowing the culture and being a modern woman, you feel strongly about not wanting to help him share this financial burden, especially as you don't actually do any fetching and carrying for her.
As I said on the first thread, his first loyalty will be his mother, then his DC then you, if the push comes to the shove.
He knows you are invested in him and your DC.
So if you are so adamant in not taking over the financial burden with your DH on this matter you would need to divorce because there is absolutely no way your Dh would ask your mil to divorce fil.
How does that scenario look to you?
Would you have enough money to support your DC in the way you would want to if you cut your DH and mil off?
I suppose the way you could go about this if you didn't want to divorce would be to separate your finances completely and have nothing coming in to the joint account for your DH to syphon off to your mil.
And then you could put your money in your parents' account so that they could be seen as paying for your DC's activities so that they don't lose out.
It would need you double crossing your DH financially like he has done to you all these years.
Protect your own money. As far as your assets are concerned, beware that if you divorce he would be entitled to half of yours including pension so this might not be in your best interests of you are the higher earner.
Anyway op, you have a lot to think about and you would need financial and legal advice of your own now that you know your DH is not wanting to work with you on the matter.
He quite plainly wants you to cough up, take on board his responsibilities to his mother uncomplainingly and stop embarrassing him.
Hope I'm wrong op.