no. If you brought it in for a no start condition, and some ditzoid diagnosed it as a starter, they installed and charged you for a starter, and it still won't start then you are entitled to a refund. I've worked for firestone in the past. They are gonna play hard to get to come off the refund but its all about pulling the right strings.
Your options, depending on the way you argue it and who you deal with, are:
a full refund of th cost of the starter, the original diagnosis, and the labor.
OR if you can play it right, refuse the refund and make them buy you the PCM.
the latter is unlikely but I've seen it done on items way more costly than an 800 dollar PCM. If anything, general shop policy is if you misdiagnose something, and perform and charge the customer for the repair, and that was not the problem, you eat the cost and refund the money. Trying to charge you for both is unacceptable.
I'm not sure of the price of the PCM, call your local dodge dealer and have your VIN ready. As for them wanting you to pay for their misdiagnosis, yes you are being had.