Obviously it's time to enlist the help of the internet for what I thought would give me a consensus. The first result is a YouTube video that gives the pronunciation of pear-a-dot. Well that was simple enough. Then I continued to scroll down to the comments (not always the safest thing to do without losing a few IQ points) and found the people there disagreeing, with some preferring the pear-a-doe pronunciation. It appeared I had some more work to do.
Wikipedia lists both pronunciations and looking to see if there was some more information I headed to the talk page. Now I don't normally frequent Wikipedia talk pages, but what I found could be called the polar opposite of YouTube comment threads. There was a fairly long and verbose discussion about the pronunciation of peridot which was strangely amusing to read. After reading through it all I determined that both pronunciations are acceptable and it can depend on where you live or if you're a gem tradesman.
If you would like to trust what was posted near the top of the talk page,
Many dictionaries put the "pear-a-dot" pronunciation first, before "pear-a-doe". Some only have the "pear-a-dot" listed, for example "Webster's Encyclopedic Unabridged Dictionary of the English Language" is very definite and lists only one form, "per' i dot'". Most gem and jewelery tradesmen use "pear-a-doe", and some are very definite in calling the "-dot" form as incorrect. If you believe dictionaries and encyclopedias, however, you have to conclude they are being near-sighted. Peridot was first mined in quantity in the U.S. in the area of Peridot, Arizona. The name of that town is pronounced "pear-a-dot".Cool I guess.