I finally got a taste of the T3 airtrain mess last week and after trying to get from BART to T3, here's what I ended up doing as someone who needed to check a bag:
Coming in — BART -> Airtrain (T3) -> Crossing the street. This took like 20 minutes. Would not recommend
Leaving — Baggage Claim -> Walk through the basement of T3 into the international terminal -> Walk to BART. Took about 7 minutes.

Needless to say, if you need to check bags out of T3 and you're coming from BART, just walk from the BART platform to into the baggage claim area of t3 and then go upstairs.