Continuous lighting makes it easier to get the right lighting placement to avoid shadows or other unexpected complications, you might not want to spend time experimenting with strobes while your child is getting tired. I find continuous lighting much easier. I am really cramped, 14 feet wide sounds wonderful, I have maybe 8 ft or less width and about 16 feet deep (A trailer). I have two 16 ft ceiling mounted flexible tracks with 4 backdrops mounted to tracks, so I can slide the one I want into position with the unwanted ones around to the side. Since I do product photography, I get by with the narrow space. I do have a full size Mannequin on a wheeled base that I can roll into place for use by my wife, who takes snapshots of her sewing, and knitting creations.
I have strobes, but purchased two low cost 27 inch softboxes from Amazon. Each has two 85 watt CFL's and two 65 Watt CFL's. I mount them to the walls with angle brackets to save space. I can swivel them and angle them, one is higher than the other. I also have a ceiling mounted light which lights the backdrop, but is not totally satisfactory. Its amazing that just a tiny tweak can affect shadows, I'd never get things totally right experimenting with my strobes. The lights are fixed, so I move the subject closer or further away, while trying to keep as much distance to the backdrop as possible.