First, that amplifier is either tiny or the Bluetooth transmitter is HUGE because wirelessly transmitting power is quite inefficient!
It's also probably extremely expensive, I've never seen a Bluetooth POWERED amplifier out in the wild. Most amplifiers I've seen are powered by the local/regional power company
{had to get that off my chest for the sake of the forum community sorry about that}
Okay, on to your question. I'm a bit confused.
What are you trying to do?
Obtain better quality audio?
Go wireless?
Generally speaking: wired > (better than) wireless
wired pros = reliable, sound quality, cheap
wired cons = cable, portability
wireless pros = portability
wireless cons = lower reliability, sound quality, expensive, digital delay
It sounds like you are saying a Bluetooth pre-amp (wireless to line level converter) [like the Radial BT-Pro] is what you are looking for. As far as your friend saying it will sound better, probably not. Though the medium (.mp3 vs .flac) is usually more impactful than the transport (wireless/wired).
What are you trying to go: TO?
Small portable all-in-one sound system?
Or Plug into provider sound system or something else?