After searching the internet and coming up empty, I asked a support rep via chat (Chad, 11/22/17 06:44:48) if TDP supports Windows trusted connections, which he called Windows authentication. He said it doesn't and recommended formally requesting it via this forum, so here I am.
He also sent me a message later (thanks Chad) saying that he'd done some digging and found some undocumented ways that may or may not work in TDP (they work in Toad for Oracle). My DBA is testing those ways right now.
Regardless of whether any of them work, it would be nice if it was an officially supported feature of TDP when working with Oracle databases.