Once you submit for arbitration, just like involving a lawyer, any progress (such as it is) will stop. I still consider it a last resort. Once the arbitration decision is made, even if not accepted, there is not going to be further negotiations and it would only be a legal issue then.
Well then KOG has 3 options:
(1) Chrysler has offered to rebuild the existing trans, which is their option under the warranty.
(2) Chrysler has made a cash offer to satisfy the terms of the warranty. It's not an option listed in the warranty that Chrysler can do that to satisfy the terms of the warranty, but it certainly is within their right to make such an offer, and they have appeared to have done it to others under the same warranty, so they have set a precedence of cash offers. This is probably the least expensive option in favor of Chrysler to handle this case. They get out of having to do any actual repairs.
(3) If KOG has retained a lawyer, his contacts with Chrysler (including his dealership) must cease, and his lawyer handles everything from this point forward. His lawyer will probably even tell him that, but if he even mentions to Chrysler Customer Assistance or his dealership the "L" word, they will immediately shut down all conversations and tell him to have his lawyer contact the Chrysler legal team through the proper channels. So in that sense, getting a lawyer is almost like arbitration, only your lawyer is doing it for you.
About the only thing a lawyer will be able to do, is perhaps negotiate a higher cash value for your vehicle, but then, if the lawyer is successful in getting more $ for you, it would be offset by whatever the lawyer charges you to handle your case, so in the end, you may be no better off than negotiating the $ directly with Chrysler.
I don't know all the details of your van so had to make some assumptions, but I used items that resulted in the highest value being estimated from KBB:
You also may be able to get a little more $ for your vehicle especially if you have done any recent expensive work or improvements to it, similar to negotiating with an insurance company if they want to total your vehicle following an accident. Many times you can get more than their initial offer.