Boots only if Battery gets disconneced and connected again

Really strange problem...

If my MacBook Air 11" Early 2015 is off and I want to boot it is only possible after disconnecting the battery once and connect it again.

Otherwise the Mac is running without any problems. Also doing a reboot works fine.

So if I shut down my mac and want to boot it, I have to open it and doing that procedure mentioned. Also using the Keyboard Power Button then (to boot) doesn't work. I have to put the power cable in, the fan is starting and CPU boots.

Any ideas? A battery change could help?

I am really thankful for any help