Depends on where you are in Downtown. The issue here is that Union Station is actually not in Downtown... more like just north of Downtown, on the other side of US-101. At that hour (~7AM), it is near the peak domestic departure time from LAX so the departure level of LAX will be fairly busy so probably take about 20~30 min from exiting the freeway to T6.

If you are anyway near I-10 or 110 freeway on ramp in Downtown LA, it will probably be faster to Uber. If you are within 1 or 2 minute walking distance to Metro station, particularly on the northside of Downtown in general, it will probably be a little faster to take FlyAway, assuming you can line up on the departure time of the FlyAway bus.

FlyAway will take a long slog from Union Station to the 110 freeway but once on the freeway, it will use the Fastrak toll lane on I-110 and HOV lane on I-105, as well as being able to navigate inside LAX in transit lane. But FlyAway will stop at all the terminals whereas you Uber will go straight to T6.

Traffic will be thick during morning commute hours. If you can, leave 30 min earlier than whatever time you originally planned no matter which method you use.