We did exactly this commute for years. I did it before Balham had the lift, with a variety of babies/toddlers in tow, so just lugged the buggy up the stairs, v good keep fit! There's no lift option at Battersea Park and those stairs are steep so do a trial run before you decide for sure. It is a squash getting on at Balham, but you need to go right down to the back of the train for the Batt Park exit anyway, and the kids don't seem to mind they just accept that's the way it is. It's a good lesson for them to learn how to behave on public transport as well. Other commuters are pretty understanding as well, as long as kids aren't bouncing off the walls. Loads of people get out at Clapham Junc anyway, so you get a bit more room then. There are lots of NP kids on that train, so it's quite nice. Another advantage is that when the kids reach Y6, they can come home on their own if you and the school feel they are ready, and that's a great safe step to independence. It made the transition to secondary a lot easier, knowing my kids were aware and confident on trains and buses.
So while you have tiny ones in buggies, it is a bit of a pain, but that phase goes v quickly and then it's a really good and easy way to get to school.