I use Lifemiles for EVA business class frequently. Normally you'll see two business class award seats far out. There's a trick to book a date in a month not yet showing up on the Lifemiles search function. Find a date not available on the furthest month out and then scroll forward on the dates you see above the calendar. That's how I got a January 2024 business class seat when the calendar was only showing December dates.

I don't worry about the domestic flight to IAH on United. I just buy that as a paid ticket closer to departure. United will tag bags on separate tickets so no worries there. Something to consider for SE Asia. Lifemiles increased the miles from 78k to 90k. You can use EVA miles and get the seat for 80k. And EVA is a Citi TY transfer partner.