TVs at that size rarely get reviewed anymore so its hard really to recommend one, especially at that budget. Because not many people buy them anymore, manufactures don't seem to place importance on that size and don't release any stand out models. Best thing would be to look at user reviews, maybe someone on here has a good 32" they can recommend.
Sony WD603 gets a good review at whathifi and other TVs like the Sony WD750/Samsung K5500 at 32 are likely average.
They just don't make great TVs at that size anymore, don't think they have since 2014.
I'm glad that you're not bothered about 4K, I would have thought that (a) 4K resolution will be wasted on a five-year-old and (b) she almost certainly wouldn't be close enough to the screen to benefit from 4K resolution. Dodgexander has, somewhere, a chart that shows optimum viewing distances, depending on screen size and resolution (SD, 1080p, 4K). Perhaps he could provide a link to it (again!).
@sub80 Eh??? What a bizarre thing to say! Even when plasma televisions were being made, 32" plasma sets were rare. Not to mention that the last time you could buy a new plasma was at the end of 2014. And, plasma sets won't have catch-up services. I'm further intrigued by your implication that LCD televisions damage the eyes - have you any proof of that?