At this point your PS4 controller should have connected in just a few seconds turn on your PS4 controller just with the PS4 center button(This is assuming you are already paired) Click on Connect Now( at this point you should see all your device trying to get connected) Bluetooth Auto Connect window open in the Advanced menu But in the other hand I think what really gets it to work is the sequence you do it in. So I thought I was missing something as well but you have to try it a few times until the lag goes away it seems like. If I discover a better solution or one is presented to me I will gladly share it with you here, so stay tuned.
Post your results in the comments and maybe together we can work it out. I call upon you with the same issues to exploit this, and hopefully we can find a steady and faster way to resolve this issue. The lag on my Xperia Z kicks in each time exactly after 30 seconds, and usually take's 3 to 5 times turning off/on Bluetooth, and seem to work best for me while playing Dead Trigger 2. I know it's not a good or permanent fix but it's all I can offer for now. until unfortunately after turning Bluetooth or controller off T_T. Once it does stick it won't lag/delay again. And when the lag/delay returns repeat these steps until it sticks. Then press the PS button on the controller and immediately after turn on Bluetooth again (controller should connect again). If after paring you encounter lag issues (delayed button input), turn off Bluetooth (the controller turn off as well) you can do this even if a game is running. The description in the Video seems to have a workaround: