Simple answer. No.
At LAX you must walk over to TBIT. Check in at the EVA counter and re-clear security. EVA has shown an anoying tendency to start wrapping up check-in as early as 1 hour and 15 minutes before the flight lately, never mind the time it takes just to get to the check-in counter.
Don't schedule to much less than a 2 hour minimum layover at LAX or you wont make it.
Interlining the luggage shouldn't be a problem though.