Your exposure is a function of -

• The amount of light reaching the subject (with the quality and direction of the light allowing you to control the effect)
• The shutter speed (too long leads to blurring as you've seen)
• The aperture (wide lets in lots of light with shallow depth of field, narrow the reverse)
• The ISO (like an amplifier dial on a hifi - controls the output level for a given input)

From the aspects you haven't mentioned, opening the aperture more will let in more light. For portraits of this sort a fast prime (non-zoom) lens will definitely help if you're not already using one, simply because it'll let you capture more of the light.

That said, I think you're likely being over-tight in your parameters and would be able to get sufficient print quality even by relaxing them a little.

• I've shot a Nikon D7000 for many years (which has a very similar sensor) and ISO 400 definitely isn't too noisy for most applications; I'd be comfortable from experience running up to ISO 1000 anywhere and ISO 5000 sometimes.
• 1/100 at a likely focal length for this sort of image is a generous minimum shutter speed; with practice of how best to stably hold your camera I reckon you should be able to get down to 1/40-1/60 reasonably safely.
• Some kids react negatively to flash, I agree, but not all and I find they generally get used to it pretty quickly. I'd practice more on the flash, and ideally get it off camera and running through a modifier like an umbrella or other diffuser to soften the light.