5th dnd – Does a kenku bard need an instrument or can he use mimicry to get one?

Your DM can not make such decisions in the Adventurers League game.

The rules are pretty clear on the components of the spells. With regard to the hardware components:

Casting spells requires special objects, specified in parentheses in the component's input. A character can use a component pocket or spell casting focus (in "Equipment") instead of the components specified for a spell. But if a cost is specified for a component, a character must have that specific component before they can cast the spell.

In other words, you need the appropriate hardware component or an element of discussion. For bard spellcasters, this focus element can be a musical instrument.

You can use a musical instrument (see the Tools section) as a target to cast spells for your bard spells.

The features of the Wizard class have a very similar passage:

You can use Arcane Focus (see Adventurer Equipment) as a spellcasting focus for your wizard spells.

Based on the published rules, to which DDAL is very close, the personal interpretation of your DM is totally wrong: aside from the permitted physical object, the bard's spell casting focus and the launch focus A wizard's spells work in exactly the same way. In fact, nothing even says that a bard has to to play the instrument to use as a center – they just need to have it on hand.

As for the use of mimicry … the voice of a kenku is not a musical instrument. Verbal components are a different category of spell component. Although there are rules for using a hand wearing focus to complete the somatic components, the character must nevertheless have the focus object.