What's the most efficient way to worsen your rank?

The fastest way to lower your League with the minimal game-ruining for other people is to leave the match as soon as it starts.

The fastest way to lower your League with no regards to the experience of other people is to leave and rejoin the same match multiple times. Pre-forming a team of 3 with everyone knowing in advance that this is the intention can help avoid any negative sentiments from this method.

Leaving a match automatically causes it to count as a loss on your record. This includes matches you've already left. The only thing that affects your rank change is the rank of the opponents you were facing and whether you won or lost. Kills, deaths, and any other statistic are not involved in the calculation at all, so playing badly will not have any positive effect. Time only comes into consideration when the enemy team changes in the middle of a match, in which case the average rank of the enemy team for the length of the match is what's considered.

This information will only be accurate up until the release of the new matchmaking servers, scheduled for Patch 3.2.

This is not an answer to the exact question, but I have to mention here another option how to get appropriately matched games with your new character: Steam Family Sharing.

With it you can create another steam account and allow it to use the games (including Awesomenauts) from your original account on up to 10 computers/systems.
Once created Awesomenauts will see you as a completely new user, you will have to skip tutorial in the options and play 1-10 games online to unlock your character (if it is blocked, like Derpl).

It took me 8 wins and 1-2 hours to unblock Derpl and at that moment I reached league 7, exactly as I wanted.
The main downside of this method is that you will need to unblock extra Upgrades by reaching max Player Level, and this will take a while.