Why would the Bellagio be better taking my daughter Simon? I thought I'd decided on PH till you've mentioned this. How did you get $434 inc taxes? The cheapest room for me is coming at $408 + $28pn resort fees, total $512. Bellagio with a fountain view is $580.16 inc everything. PH with fountain view is $355 inc everything.
I got that price following the clicky I posted ^^^ and afaics included RF, tax etc.
PH is a very vibrant hotel, aimed at a fairly young crowd that came to party. You can walk your daughter to the buffet, pass through the pleasure pit and watch the scantily clad go-go dancers dancing on poles and into the buffet... The pool is a small slab of concrete, massively over-populated with loud music and drinking.
Bellagio the pool area is massive, and (generally) very sedate. You will easily be able to find sunbeds, parasols, and there are at least half a dozen pools, hot tubs, plunge pools, etc etc for you to relax and enjoy.
There's no right and wrong answer. I like PH, I wouldn't put anyone off staying there, far from it. Definitely check out Elara too which might give you a decent compromise ~ if you can somehow get a 1BDR suite for similar to Bellagio, that would be a solid family choice too.
It all comes down to personal taste and propensity to part with $$ ~ as long as you know what each experience is going to be like, you can't really make a bad choice.