What bothers me about continuous lighting rigs is that they're a bit "squinty" for your prospective clients to be staring past whilst you get your shot ready.
There's always the issue that flash might upset the baby, but I think it would be easier on the adults.
Flash with modelling lights would be best of all - but that's not going to be cheap.
I often actually use both together, continuous & flash, trying to mimic modelling lights [though not precisely as I can't get them both into the same position] but the continuous lighting helps for speedy focussing. I also have adjustable brightness continuous lighting, so I dial it back as far as it will go to overcome the 'squinty' aspect.
Of the two choices, umbrella vs softbox, I like softboxes for tight portraits, maybe umbrellas by the time I'm wide as your family shots, but I don't think it's the make or break choice. Close to a back wall like that, neither will kill your shadows on the wall.
If you used either of your proposed continuous lighting solutions & hid them behind large white sheets, you would spread the light & simultaneously reduce the 'squint'.
Hang or frame a sheet at 45 - 60° both sides of your space & put your lighting behind it so it illuminates the whole sheet. [drawing only shows the far side, nearside would mirror this]
Excuse the dodgy artwork ;-)