"What method have people used in the studio for a good mellow sound? - this is a wooden ended duet. Two mikes? Type? Like to get a good captured sound ( not everything can be fixed in the mix )"

Several things here. One is that that judging things on what they sound like on their own in an ensemble is not necessarily the way to assess them – you might find that rather than needing 'fixing' in the mix, that very perceived harshness is what makes the concertina sit well in the mix without requiring any EQ. I would wait until you hear it in the mix with everything else before you decide it sounds harsh. Maybe you already have?

That said, a smooth large-diaphragm condenser mic such as an AKGC414B or 214 will sound good. Or, if using a small-diaphragm condenser (pencil mic), back it off the concertina (assuming you've got enough separation to avoid bleed from other instruments).

Someone suggested not using an SM57 (or was it 58?). I wouldn't necessarily agree: I think positioned in the right place an SM57 or 58 ought to work.

If he has an electro voice RE20, a Shure SM7B or any ribbon microphones, that'd give a warmer sound.

Using more than one microphone might result in a better sound or it might not - depends which microphones you use! I wouldn't personally use two mics on it unless it was being heard solo: if you are an ensemble with vocals and strings (more than one?) as well personally I would want to leave as much space as poss in the mix for the vocals.

If the vocals sound good, then that's 90% of your mixing work done; everything else is subservient to that if the songs are worth hearing. What does the vibe of the concertina recordings sound like? If the playing is good and the energy and vibe is right on the concertina takes, I wouldn't re-record, I would EQ out the harshness. It's easy enough to do: that's what EQ is for! However, if your squeezebox player is unhappy with their performance, you may as well use a more laid back mic when you re-record.

Got any sound files to share? Very difficult to give advice like this without being able to hear what you're talking about. Are you 100% sure that the monitoring system you're listening to yr raw recordings on is accurate?