No one can force anyone to settle a case by lump sum settlement (compromise and release--C&R). That can only happen if both sides are willing to settle that way. Otherwise, the only thing that can happen is to determine the percentage of disability (either by agreement or by trial), and get an open medical award. That would be why your attorney proposed settling by stips.
Since you have a lawyer, you really should be talking to him about this. He knows you and knows your case better than any of us here on avvo.
There is no formal time line for settling cases. To settle a case, you have to have a willing seller (you) and a willing buyer (the insurance company) and you have to agree on a price. You know from buying a house or a car, those negotiations can go on for 5 minutes or for weeks and even months (look at how long the negotiations in D.C. just went on). You've already pointed out one glitch in the negotiations. Since your attorney doesn't get paid until he gets the job done, he has an interest in getting the case resolved as soon as possible.
It is so hard to say without more information. Perhaps your attorney can file for a status conference at court to push the carrier to produce the copy of those checks and try to move toward settlement.