Still I don't understand why your new company's lawyer askes to speed up the old H1 extention, in order to compensate the gap? Usually you start the H1 transfer first before you tell the old company to leave. And I think new company can file H1 transfer even you are in extension pending state.
And, if a premium process can get over all these trouble. I don't think the old company has to pay for it. It is not their business as long as you have left them. They are responsible only when you are in their company.