In theory you have a good idea with the child's size bike, but in practice, and for most underheight people, the 'child's bike' idea isn't likely to be as useful as most people might think - or as we shorties ourselves often hope!
The problem is that adults - whatever their height - are of different general proportions and weight distribution to children. Also, an adult of the same height as a child will usually weigh considerably more - from bone and muscle mass alone, even if there's no factor of overweight - than a child. So a child's bike could present just as many fitting and comfort problems as an over-large adult one, with the additional issue of possibly exceeding a lower weight limit/invalidating a warranty, especially if the rider is sturdily built and wants to carry the weekly shop or camping etc gear.