I have done it. My workflow in opposite to your suggestion: I have copied only the corresponding OTA files to SD card and have let the switch done by the updater App. It worked seamlessly.
First: if the reason for the switch is to get on Android 6, then you should note that the next update of FPOOS will be a Android 6 as well.
Second: if you want to proceed anyway, I would suggest to first switch back to the last FP OS 5.1 (with the official OTA-switch file; i.e., by using the updater app), and then simply upgrade your FPOS to the latest version, hence upgrading to Android 6.