1996 Infiniti I30 Back-ordered part problem

I need some advice regarding a repair. I had my car in the shop earlier this month for a check-engine light issue. They told me that the check engine light was indicating a speed sensor fault and also a bad knock sensor. They quoted me a price to repair and I said go ahead with the work. I also asked them to perform a Comprehensvie Service Inspection, as I was preparing to send the car to my sister in Florida and wanted to know if anything else needed repair. They performed this inspection and recommended new front brakes, which I also approved.

The repairs took almost a week to complete "because they were waiting for a part to come in". They finally called and said it was ready. I went to pick it up, paid and left (this was June 5th). I then made arrangements to ship the car and it went on a truck to Florida a few days later -- it's there now and my sister loves it.

I got call last Thursday (June 19th 2 weeks after picking up my car from the shop) -- "that the part came in" and to call them to schedule to have it installed. I said "what"? "I didn't know anything was back-ordered for the car". I was confused and had my sister fax me the invoice (I had shipped the maintenance records to her).

On the invoice they charged me $298.99 for a part and also $93.50 for the labor to install it, but I wasn't sure it was the part in question or not. So, I took my copy of the invoice to the shop to talk to them. By the way, there is no note on the invoice that anything was back-ordered. The guy I spoke with was surprised that they had already charged me for the part and also that they had charged me labor. They showed me the part -- it was a whole speedometer head (dial) replacement. I said why would I need that -- the speedometer/odometer is working just fine -- I thought I had a 'sensor' problem. He said the check engine light indicated that the speedometer head (whole dial thing) needed replacing. I said my sister just had to have the odometer verified to title/register it in Florida -- the odometer is working. He was perplexed at this also. He told me I would have to come back when the Manager was in on Monday. I stopped by the office on Monday and the manager was out. So, today I just received a voice mail message that I should come by the office again and they would refund the labor cost and I could pick up the part that I've already paid for -- meaning they are not willing to refund me the cost of a part that I can't do anything with at this point.

So, my question is this -- they never verbally told me anything was back-ordered # 1) when they called to say the car was ready and # 2) when I was standing at the counter paying and # 3 the invoice doesn't state this either.
I was completely unaware that any work was outstanding -- otherwise I would not have shipped the car to my sister in Florida until everything was complete. Am I obligated to pay for this part? Even if I pay for it -- how could my sister even use it since the odometer reading that has been 'set' would basically "roll-back" the odometer (right?).