You would benefit from having a MUCH larger reflector, to REALLY bounce some of the window light back onto his shadowed side. I'm relatively confident, based on nearly 40 years's of photography experience, that the reflector is doing at least "something" (and likely a "fair bit of something".)
One of the single biggest mistakes/failures that I see on web-based lighting diagrams, and even in videos posted on-line and made by novices, is that the actual, diagrammed position of reflectors is, quite often, flat-out WRONG. Just flat-out, uselessly, ignorantly "wrong". No other word for it,sorry to be so plain-spoken. And I am not saying that is your issue here, but am just pointing out that, as a person looking for answers, that quite often in todays's web-based photo instructional world, you will encounter videos, or group discussions where novices (say 6 months to a year's worth of experience) will"show you how to, " _____________" with lighting equipment--and they will give you bad advice, or show you flat-out "wrong" methods to light!
The single, biggest problem with using a window light setup is that the light output is usually very concentrated, into a SMALL area, that spreads out in kind of the shape of a basketball key. To efficiently reflect that weak, and diffused light, the reflector needs to be quite large. And by that I mean "think at least three times the area of the window". Think ,"queen bed sheet, stretched tightly on a PVC pipe frame". Think BIG REFLECTOR, if you really want to effectively use diffused window light from "normal-sized" windows, found in most homes. Look on-line for DIY plans for making a PVC pipe lighting diffuser. A pipe frame, with some legs, will allow you to set up a BIG reflector that will really work, and work withoiut a lot of hassles, and without much need to re-position when the kid moves a little bit. A reflector that is say 24x 36 inches is, frankly, not that easy to use without a LOT of constant re-positioning and ultra-careful attention to detail.
You want a set-and-forget kind of deal...so...think "twin bed sheet--or BIGGER reflector." Or, think four, large, poster-sized pieces of foam board, attached to a Walmart rolling laundry stand, to make a "rolling flat".