The Castagnari Lilli can be very noisy!
The old Hohners have a nice quiet mellow tone.
Try lining the bellows with sound absorbing material, (maybe just fill with a lump of rag) and putting a baffle inside the RH grille. Some concertinas have a leather baffle for this purpose. For the bass end, cover most of the holes in the LH cover. You need to leave room for some air to get through!

You may get somewhere by acoustic absorption in your room too. If the sound is bouncing from wall to wall you're getting a lot more of it than you need. Otherwise find times when your neighbours aren't in. Or when they're playing loud music anyway. (been there, done that)

Anahata