There are problems with the last two solutions: Range. BT max range is about 30 feet so depending upon the distance (plus walls and flooring) between kitchen (w/speakers and Dot to transmit) and downstairs (w/amp) this could be a problem. It likely is.
The best solution I believe is have a Dot in kitchen and a portable BT speaker to take to kitchen, or get a Tap (w/ Alexa and BT, and stereo speakers) which is why I also have a Tap, Dots, and Echo.) Alexa and music can be wherever we want it.
Recently bought a 6-pack of Dots; three will be gifts this month, and 3 others are in addition to the two we've had. We are never far from one of these little buggers. Two new ones are permanently placed somewhere handy, and the last new one is a "floater" for wherever it may be needed. it is also our travel Dot that already went with us on a 10 day vacation at Thanksgiving time! Only needed to connect the wi-fi at new location and change location so we got the local weather report. (Just don't forget to change it back to home settings when you return.)