@the500 Wow I can only imagine how frustrating it was that you got the BOCA-authenticated marriage certificate and then moved to a different TECO which then wouldn’t accept it.
The TECO NY instructions also say 『結婚證書或國內最近三個月內核發載有結婚登記之戶籍謄本正』 i.e. marriage certificate or household registration transcript, but when I emailed them they insisted that I would need the household registration transcript, so I was pretty confused.
But I think I am actually starting to understand why this is so confusing. The TECO LA Chinese instructions are actually much more specific: 『戶政機關核發之結婚證書或國內最近三個月內核發載有結婚登記之戶籍謄本正』 i.e. a marriage certificate issued by the household registration office OR a household registration transcript showing the marriage. That is, one way or the other, the paperwork has to come from the household registration office. That is, the marriage certificate that your parents and my parents had that was issued by a district court in Taiwan is NOT what they mean when they say marriage certificate.
The AIT website on marriage leaves a few more breadcrumbs to follow:
Also effective May 23, 2008, marriages performed by Notaries Public at a District Court are not final legal marriages until they are registered at the HHR Office. In other words, couples who use the District Court must also register the marriage at the HHR office in order for the marriage to be completed under Taiwan law. The Notary Public can only issue a “Certificate of Notary Public,” and cannot issue a “Marriage Certificate.” The official date of the marriage under Taiwan law will be the date that the couple registered it with the Household Registration Bureau, not the date that the couple appeared at the District Court.
So since 2008, marriage certificates come from the household registration office and so therefore are essentially the same as household registration transcript. Before 2008, it was possible for a marriage certificate to be separated from household registration. I kind of think this is the root of why there’s so much confusion around the marriage certificate.
I feel like in theory this should allow for an argument that a pre-2008 marriage certificate should be acceptable, especially if authenticated. In practice, I don’t know whether it’s actually possible to make and win an argument at TECO. 
In practice, both option #1 and option #2 are probably similar in difficulty/cost/time for me so I am thinking of pursuing just option #2 which seems like it has a higher likelihood of success, and also seems like I should get my household transcript properly updated anyway or there will be trouble further down the road.
Thank you for your advice and sharing your experience!