Installing TWRP on a locked OnePlus One without wiping data
NO. Period.

If you want to unlock the bootloader erasure is inevitable. You can make a backup, but I'm not sure it'll be restored. However what is the point when you flash a custom ROM you'll need to wipe data in that case anyway. However you can move media to SD card pull it out and unlock bootloader (Make sure the SD card isn't encrypted).