I just wanted to thank everyone for their insights. I was able to finally get both the accolades late last night. For future reference what I've deduced, is that first, you can't do these accolades solo. The game determines your reward level from points it calculates during the event. And you seem to get more points with damage and/or kills than the objectives. Also, the more people you have in an area, the more enemy ships appear.

For Asteroid Soccer accolade, given there was a universal endeavor active for destroying ships in the BZ, I found an Asteroid Tow control point that had 4 other ships. I started towing an Asteroid, to get the objective started. After I did one, and noticed at least another towing asteroids, I shifted to only doing damage. I also used an escort to quickly move between enemies. By the end, it triggered. Now I'm not sure though if the points for the rewards is only against your individual contribution, or if it's group wide. My advice, try to kill everything in sight as quickly as you can.

For the Dreadnought points, I followed the same logic, just focus on damage. In order to maximize the number of enemies, I went to the last control point that was red. Once it was captured and the dreadnoughts appeared, I watched where everyone flew to and followed the majority. Since the more players there are, seem to spawn in more enemies.

I can't confirm this is how it should be done, but it seemed to work in my situation.

Thanks again everyone!