Oddly my update to your comment didn't seem to post, which is too bad because I included a lot of details.
Your tip was very helpful - after removing the config files and re-sharing the drive I was able to load it up as a second storage device. I inventoried and catalogged and this time, when I started a new Exchange restore job, the older backup sets were visible (yay!).
I started a restore job for just the single mailbox I need and unfortunately it threw an error related to credentials:
"The job failed with the following error: Cannot log on to EWS with the specified credentials. Review the resource credentials for the job, and then run the job again."
I checked my share first to confirm that all the permissions were clean - confirmed that Everyone had full access, but just to be safe I added the network admin account I use to manage the server as well as the backup exec System Account ("system-backup'). I drilled down into the advanced options and added both to the Permissions table there as well.
Back in Backup Exec I ran "Test Credentials" and all the results came back "Successful".
I tried the restore again, and this time I got a different error:
"The job failed with the following error: A communications failure has occurred between the Backup Exec job engine and the Agent for Windows."
I feel like I'm really close, but I am at a loss for how to proceed. Any advice you can offer would be greatly appreciated!