It sounds like your sister is envious of your life, especially after splitting up with her ex and going to live with your mum - that's a massive downgrade. When your mum died, it was hard for you all, but your sister may have been holding onto her as anchor without realising.
You said your mum was all about family and it sounds like she may have been enough of a presence to hold it all together, in spite of sibling rivalry etc.
Inviting her out or to your place is not insensitive if the intention is good from your side.
No contact is hard, very hard and is not to be undertaken lightly, it is literally the last resort and not something you can't really go back on. Ever. You also have to fight the evolutionary biological drive for reconnection on the regular - forever.
When you choose to go hard no contact, you're essentially choosing in the present moment never to hear from or about them again, or go to their funeral
I've been through the whole years of drama because I bothered to get my shit together and jump out the fishbowl. Everything I achieved or did was apparently rubbing my family's face in it, because of what they hadn't achieved due to fear. I got 'bored' of having to live small in their presence to keep the one-sided 'peace'.
My family of origin was fucked-up a long time before my last parent died. I chose to go no contact (& about 10 years too late) 4 years prior to my surviving parent opting for slow death by what was by all accounts, a brutal and undignified descent into severe alcoholism that robbed them of all agency.
No contact may not be the right move for you, it was for me because the destructive patterns went back 2 decades, rather than any sibling rivalry or envy becoming apparent after a family tragedy.