The workaround is to trick Thunderbird to not pay attention to that new notification.
0) First, make sure you're subscribed to "All Mail" if you (like me) thought unsubscribing would solve your problem
1) Create a filter to mark new items in "Sent Mail" as read. Tools -> Message Filters -> select the gmail account -> New
- apply when: checking mail or manually run
- match all of the following: [from] contains [email@example.com] perform these
- action: [mark as read]
This way Thunderbird won't think they're unread. However, at least on my machine the account name still highlights blue and a notification sound is played (perhaps because it can't run the filter fast enough?) So, step 2...
2) Right-click on "All Mail" -> Properties -> General Information
- UNCHECK "When getting new messages for this account, always check this folder"
- (you probably also want to uncheck the other box, "Include in search results" so you don't have duplicates in search results, but that's a different issue)
Send a test message to yourself or some bogus account to test it, but it should no longer give you a sound notification or turn the account name blue.
EDIT: I discovered that after doing this, new received items still show up twice (once in inbox, once in All Mail), which is suboptimal. I found that simply unsubscribing from All Mail after performing the above two steps solved this last problem without affecting the sent mail problem.