It is possible that there is no award space. The middle of December is generally high season. I would keep checking though.

I just checked on December 9 and there is 1 seat available in biz for BR 15 and BR1 from LAXTPE. There are multiple seats for BR11 in economy.

The aircraft rotation for Eva Air's Hello Kitty flights also isn't posted past November 2, 2014. Should also be noted they don't have daily flights on the Hello Kitty Jets from LAXTPE. You might consider going via NRT on UA and then connecting to TPE on BR. This will save you some miles and there are daily flights on the Hello Kitty jets from TPE. You might try routing LAXNRTTPEHKG as you should be able to get the Hello Kitty flights on NRTTPEHKG.

If you want the full experience I suggest you take a flight ex-TPE as I believe TPE has all the bells and whistles from check-in to the gate area.