You need to list the positives in your life and if something is causing you stress, you have to determine the change that you need to make. Sometimes it's as simple as working to change your attitude - it's really the only thing we can change in ANY situation.
The way I see your positives: Great husband, wonderful son, mother who watches your child full-time, spoil them rotten, saving money big time on child care! ...etc....
The negatives: Their critizism and negativity, lack of mom/sister/grandchildren relationship (not your problem!) inept diaper skills...etc...
Would you have to stick this situation out 2 more years - you are already having angst about it???? Sounds like you benefit, BIG TIME, from your parents involvement in your lives. There will always be give and take.
As far as mom/sister relationship - and who was the favorite and who was not.... that's all in the past! My life, in a nutshell: My husband was 4 of 5 in his family - always the really good son but number 5 in the "favorite" category. Yet, we were the ones who opened our home when my mother-in-law couldn't live alone anymore in her home. We took care of ALL her personal needs, NO ONE BARELY VISITED HER FOR TWO YEARS-called, but, couldn't even come on Mother's day. WHY???? Because of their twisted jealousy of Mom having a new favorite! HUH????? We worked to remodel and sell her home...etc... all the while, we listened to her dail phone calls from her "other children" At the end, before she died, I believe she did have a deep respect for my husband/her son.
My point..... you can invite all of them to your house as a nice gesture and a neutral spot (doesn't matter if you live closer/she lives farther!) however, if your sister resents her relationship with her mom, (and resents you possibly for moving into the "favorite" spot), she needs to work to repair it, not you. Don't get yourself in the middle of it and should your mom begin ANY negative discussion regarding your sister abruptly end it. Keep your conversation positive. Maybe she feels your sister doesn't make time for her??? You're enabling the situation by letting your mom dump on your sister, via you. You may be concerned about getting your mom upset and you may not be the favorite by taking this stand but it's the chance you take for doing the right thing. If your mom doesn't have any outlet for negative talk, she'll either find someone else to share her negativity with or actually have a conversation with your sister and your sister could initiate it. Don't continue to make this your problem. Life's too short and your sister needs to mend this, if this is her desire. The won't be around forever.
Lastly, if your son needs speech/social evaluation, go to your local school district. At three years old, many districts offer these evaluations free - you just have to make that appointment and schedule it. Whether you believe that he needs any help in any area, it's good to get the evaluation and may help with determining the path that you want to take for a child care change/or not. Good luck.