Solved! I want to pair the CD-playing little shelf stereo in my mom's dining room, to Bluetooth speakers in her living room. I have a

Solution
Even if everything is compatible, I doubt a Bluetooth transmitter will work across the house. Before buying anything new, make some tests: Pair the speaker with your phone, leave the phone on the shelf with the CD player, and move the speaker to the dining table.
If the shelf system doesn't have a record output then you can connect a speaker to line level converter to the same terminals as the speakers. A BT transmitter can connect to that.
 
Even if everything is compatible, I doubt a Bluetooth transmitter will work across the house. Before buying anything new, make some tests: Pair the speaker with your phone, leave the phone on the shelf with the CD player, and move the speaker to the dining table.
 
Solution

I agree! About 15 feet is the maximum range for bluetooth connectivity.