As long as you don't dislike the firm that your LTD uses to try to help you get on SSDI, I'd be inclined to use the help that they provide.

I have to assume that the LTD company won't be too pleased with them, and would not be likely to use them again, unless the lawyer/lawfirm/non atty rep makse sure you get approved for SSDI. LTD doesn't want to pay you forever. If the firm they choose gets the job done in a time period that the LTD can live with, I'd guess that would be about the best you could do.

If you hire a lawyer/firm on your own that lawyer gets 25% of your earned backpay up to a cap of $6000. If a lawyer is working for you and you are found to be totally disabled and qualified for SSDI fairly quickly, say with only 2 months of backpay owed on a $1000 a month SSDI case, that lawyer would be paid $500 (25% of the total $2000 backpay amount) and you would get $1500. We have to assume that most lawyers would prefer a $6000 payday per case over a $500 payday. It would take an evil person to deliberately sabotage a case at the early stages, but you can see how a that lawyer has no financial incentive to try to push a case through quickly.

I think the firm recommended by the LTD would have more incentive to try to get the job done quickly.

You probably don't feel "lucky" about anything right now, but it is HUGE that you are already covered by LTD. At least you know you have the proof that you are unable to do your previous job down pat, now it's just making sure the medical documentation proves that you can't do any SGA. CRPS is tough enough without having the stress of total loss of income. I hope the SSDI process goes as smoothly as possible for you.