Thank you all for your help and advice.
I've dug around some more and wanted to clarify a bit and also add in additional 'wrinkles' to the problem.
1) Issue happens regardless of use of a headset or handset. I've also ruled out phone models, it always comes down to whether the DID extension is on our copper or fiber line.
2) We're actually running two 3300's in a cluster. Both with the same release and software load. On unit one: module 2, that has our fiber line (via a T1/E1 W/Dual Inband Loopback), both ports are plugged in. I checked the COS and auto answer is enabled. (Just for my own edification, to check a trunk line COS: I would go to Trunks>Digital>Digital Trunks>See what Trunk Service Number they use. Then go to Trunks>Trunk Attributes and look up what COS is assigned to the Trunk Service Number used.)
On unit two: module 1, that has our copper line (via a T1/E1 W/Dual Inband Loopback), only one is plugged in. I checked the COS and auto answer is enabled. Strangely enough, the trunk service number for the trunks is different than the fiber lines on unit one, but regardless the COS referenced is all the same. Also looks like we're not syncing that page between our units, not sure if that even matters.
I'm at a loss whether this confirms or busts my feeling that it's configuration issue vs. an issue at the provider end. I'm wondering if there is a way to trace a call within Mitel (I came from an Avaya shop at my previous job, so please excuse my n00b-ness) on each of the units while I make a test call with auto answer enabled just to see if there are any differences.