No doubt that the Developer has the right to demand clear payment before delivering vacant possession of the Property to you, so let's get that out of the way first.
As for the late payment interest, it's not unusual for everyone to deny responsibilities. You can verify the fault yourself by doing the followings:-
1. check with the developer which stage of the progressive billing was late; and
2. counter check with the bank and see when they disbursed the progressive loan sum to the developer.
From there, you should be able to work your way out.
Unless the late payment interest was incurred on the first disbursement, high chances are that it is not the solicitors issue because after the first advise, the solicitors basically do not need to do anything as the developer will issue their progressive billings directly to the bank and the bank will issue payment accordingly.
My take, it is the Bank's fault or that the Developer stupidly sent the progressive billing late or something like that. There should be acknowledgement of receipt of everything.
Get the bankers to have the bank issue a chronology of disbursement.
Normally it is advisable to ask developer to give their prove and chronology event
But who will take the blame is the problem, lawyer/banker or developer themselves ?