If you are concerned about having adequate lighting, I would put in a fan without a light and install separate light fixtures (perhaps two 4-inch recessed lights in the shower).

Unfortunately, there is no attic (it's a flat roof) and opening the window has limited benefit because, given the layout, the fan also ends up sucking the air from the window instead of the shower. The house is in Northern California with mild climates so I would prefer to put in a fan that is overkill (not worried about the chill factor) but the most powerful one I can find to fit 2 x 6 joists is 110 CFM. Thanks again for all your help.

Brass, particularly in a satin, raw/natural or aged finish is back in a big way. (I'm using Delta's champagne bronze finish [looks like satin brass] in my bathroom.) But if brass in not your thing, chrome is a classic and probably the best option for a kids bath. It's also less expensive, more widely available and easy to match between brands. Your photos don't include the lighting, but it's considered acceptable, even preferable, to mix metals, if you don't want to replace them with chrome. Would you consider having your contractor add a recessed mirrored medicine cabinet over the sink instead of just a mirror? It's more of an investment than an off the shelf mirror, but the storage is priceless.

Fan ceiling placement, fan size, climate (dry, arid vs humid), exhaust duct size/length/bends, exhaust wind exposure, quick vs long shower, temperate vs hot shower, there are too many variables. And, an 80-110 CFM fan is small, it's maybe OK for a powder room. I take a temperate, quick shower and I fog the bathroom every time, and that is with a 150 CFM bath fan just outside the shower, an in-line booster fan in the basement, and the bathroom door cracked open. Unless you have a large bathroom and a high capacity exhaust fan, I see fogging the mirror as normal.