Is the split straight with nice clean edges, or is it jagged with ragged edges? Bettas often tear their fins which results in splits with straight edges. These heal by themselves.
Are the edges of his fins ragged as well as blackish? Black can be a sign of finrot but also part of a betta's colouring. Betta colours are quite complicated with a lot of genetics behind it. They can have a scale colour with a different colour underneath. Blue bettas often have red under colour. And the fin style butterfly has been developed where the body is one colour and the edges of the fins another. In tricolour bettas, the body and the fins next to the body are one colour, the middle of the fins are a second colour and the edges are are a third. Moonbeam is not a butterfly but he could have butterfly genes causing a black edging, if it is not finrot.
Clear edges to fins are new growth. They do colour up later.
So, now I will suggest keeping up the daily water changes and wait for the Myxazin. If the fins start to look worse, then yes, add more eSHa 2000 until the Myxazin arrives. But unless the fins do change for the worse, just water changes.