best way to travel from Lyon to Paris with 2 infants and 4 adults

Man at Seat 61 says that most Paris trains start in Lyon at Perrache station.
Paris will almost certainly also be a terminus.
This should give you time to manhandle bags and babies at both ends.
https://www.seat61.com/international-trains/other-cities/trains-from-Lyon.htm

The train takes 2 hours, five hours minimum to drive,

Keep in mind if renting a car, you will need to 2 car seats, also. So larger vehicle required.

For train, SNCF Luggage Delivery Service, may be an option. I have no experience with it.

How much/size bags and paraphernalia do you expect to have? Are the adults all able to assist with bags?

Just remember that the fast trains cover the distance between Paris and Lyon much faster than a car can.

Train for sure! Far fewer hassles than by car, no need for car seats, no one-way rental fee to pay, room to move around if babies get fussy.

Train.

Put the babies in snuggly carriers so all hands are free while boarding the train.

Just weighing in to correct some misinformation. Above is stated:

Man at Seat 61 says that most Paris trains start in Lyon at Perrache
station.

That may be true as far as "Man at Seat 61" may indeed have written that, but the part about Perrache is simply incorrect.

As a former Lyon resident who used to take the train to and from Paris for both business and leisure, that didn't ring true. So I checked SNCF Connect and my surprise was validated. None of the train trips from Lyon (generally, no specific station) to Paris (same) originated at Perrache. They all departed Gare Part-Dieu in Lyon except for a few that departed Lyon Saint-Exupery TGV (adjacent to the main Lyon airport).

Edited to note that, in a way, the statement about Gare Perrache could be true, in that you can get on some of the trains there before they head across town to the main train station at Gare Part-Dieu to pick up the majority of the passengers before departing to Paris. Think of it as an overpowered, large street car for the trip from Perrache to Part-Dieu.

But SNCF Connect still lists them as leaving Part-Dieu for Paris, not Perrache.

None of the train trips from Lyon (generally, no specific station) to Paris (same) originated at Perrache. They all departed Gare Part-Dieu in Lyon except for a few that departed Lyon Saint-Exupery TGV (adjacent to the main Lyon airport).

That's because SNCF Connect defaults to Lyon Part-Dieu if you just enter "Lyon" as a starting point. But you can count on roughly one train per 1-1.5 hour from Lyon Perrache to Paris on a weekday. More convenient than Part Dieu if you are staying on the Presqu'ile (e.g., near Place Bellecour).

Interesting. I'll look into that to see if what you've mentioned is accurate. My office was a short walk from Part Dieu so that was my default, but I also know that SNCF's website (which predated SNCF Connect) focused on Part Dieu for arranging tickets to Paris.

Perrache, which is under renovation (as is Part Dieu, incidentally, as a much larger civic improvement) is a rather decrepit train station and is far less busy than Part Dieu.

What Balso said. If you search and specify Lyon-Perrache, you can see what’s available.
Best choice would depend primarily on where you’re staying in Lyon, but if the train originates at Perrache, you might have more time to board. Note that trains that go to Part-Dieu and not Perrache may originate in Marseille (or somewhere else south).

Being less busy is probably a significant plus for OP and her infants. I found Part-Dieu extremely crowded, made worse by arriving too early -- 20 minutes before departure (when track numbers are announced) would have been ample. Literally not an open seat in the place. Apparently SNCF has a first class lounge there but I didn't have the right ticket...

Note that Trenitalia (trenitalia.com) also serves the Lyon-Paris route (possibly only Part-Dieu, not sure). AFAIK the service is very similar to TGV Inoui.

Trenitalia serves both Perrache & Part-Dieu, but has fewer trains that go to/from Perrache.
If it is cheaper than the TGV (it often is), it is an excellent option. Just as fast, and most seating is in bays of 4 seats facing 2-2 which would be good for your group.

There are many families who take the train with their children every day!
You can choose to be in the lower deck, it's a little be more easy to handle.

FYI the waiting room at Part-Dieu is for anyone, not just first-class ticket holders. It is indeed a crowded station. Seats in the main area have to be spotted and grabbed. I've always found seating in the waiting room.