What does your solicitor say? How come there is no charge? What happened when you bought your first house that there is no charge?
It seems to me that you owe AIB €340k for which they do not have security.
That does not mean that you don't owe them the money.
If you don't sign the charge, and you are in arrears, they will get a judgement against you.
They will then register the judgement as a judgment mortgage, so they will in effect have a charge anyway.
If you sell the property when they have a charge or a mortgage against it, the charge or mortgage would have to be paid first. You will still owe them the balance and they could get a judgement for that and lodge it against your family home.
They won't want all the legal messing about, so they might do a deal with you. Offer to sign a charge for them on condition that they agree to waive any shortfall if and when you sell the house. I suspect they will refuse, but you could try it.
If they have no charge on it, and you go for a Personal Insolvency Arrangement, the PIP might well regard AIB as an unsecured creditor and apply the sales proceeds of the house against your AIB loan and the shortfall on your new home. AIB would veto that.
If you can make full repayments on €540k, you could probably reduce the capital element of that repayment and make payments to AIB.
First - ask your solicitor - probably not the one who did the conveyance for you.
Second - meet with AIB and try to do a deal.
Third - if they refuse, then go to a PIP.