My suggestion is to invest in a good portable headset. Someone on FT recommended this one when I was looking for a good one:
https://www.amazon.com/gp/product/B00DQ5NU76/
Any particular reason why you want to go with a Bluetooth speakerphone and not a headset? Only because if it were me, I'd easily forget the speakerphone was on the visor when I dropped off the rental car. I've done before with other items, like rear-view mirror parking tags and sunglasses.
EDIT: By the way, the headset I recommended works for 2 phones at the same time, if that helps.