No you don’t need to delete your Funimation account in preparation of this.
From the 17th of June you will be able to link your AnimeLab account to Funimation. If you already have an existing Funimation account there will be further information to come, so stay tuned! This does however mean that if you have a current AnimeLab subscription you will be able to use this on Funimation from this date once your accounts are linked.
At this current point in time your accounts won’t be linked, however stay tuned for more information to come about being able to log into Funimation from your AnimeLab account. As you have an existing account on both we will have some further steps on how to link the two.