Three Sonos Amps with ceiling speakers may be price competitive with other products as well like a Yamaha RX-A6A receiver which has 3 zones with MusicCast built in. Sonos is a good option though for their long term support and reliability.
On a far less expensive side, you could consider a dedicated multi-zone amplifier, used, off eBay, which is designed for this type of work.
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Then pair with with three of these, one for each room...
It's about $500 for the amplifier and another $450 for the streaming devices. Call it about $1,000 total, plus speakers.
Please keep in mind, you really don't want to use Bluetooth as that is super distance limited and requires pairing and setting up all the time. With something on WiFi you can just set it up and then jump into the apps to adjust the different zones for playback as you choose.