Backup and restore without root?

I'm currently running Android 6.0 DP3 on my Nexus 5 and have to perform a clean flash due to battery drain issues. However, this will make me go through the whole backup and re-setup process again - backup images, backup Whatsapp, backup all documents and download, flash the phone, restore everything, log into the different apps, set up Gmail accounts etc. Obviously, I'm not looking forward to that :D it's basically also the reason which has kept me from trying different ROMs on my phone in the past. I know that there is Titanium Backup just for that job, but it needs root access and I can't root my phone because this will trigger the "phone isn't secure"-alarm of my bank's super cautious TAN app which will stop it from working (just like using Tasker does, which obviously sucks).

So I am looking for a way to backup my phone's data, apps and app data without having to root the phone. Is there any possibility for that?