D&D 5E is designed as a game for a cooperating party of heroes. It isn't particularly designed to model this kind of thing. Even leaving aside PvP, the rules really aren't optimal for single PC characters acting alone.
That aside, though, some points on your plan. Particularly, you're making it much easier for the player to hide in combat than I think is supported by the rules, even using facing.
Breaking line of sight under full cover hides your location. You are hidden. your location is unknown to player in question.
This is not the case. See the sidebar on Hiding on p177 of the PH. Being unseen is required to attempt to hide, but doesn't mean you are hidden by itself. The sneaky rogue needs to actively hide.
While hidden, to do a sneak attack, the player has to approach from behind (or slightly off) and succeed on a stealth check equal to or above the targets passive perception every turn, or be noticed. If multiple opponents are in the vicinity, then unless they are facing the hidden character, the check must be greater than or equal to the highest passive perception of said players.
Ofcourse, D&D 5E has no concept of "sneak up from approach from behind" in combat by default, because all characters are assumed to be actively moving and looking around in all directions while fighting. By the normal rules, your sneaky rogue will not be hidden once they leave an actual hiding place. (This is the quote you give in the question.)
The exception is given on p177 in the PH, which says
However, under certain circumstances, the Dungeon Master might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack before you are seen.
Just looking the other way isn't "distracted", and probably in fact "in combat with someone else" shouldn't count either (given the general assumption of high situational awareness in combat).
You're using the optional rule in the DMG (p252) for facing, but this doesn't work the way you seem to expect. Particularly, it doesn't say anything about perception, so the basic fact of "leave hiding place, probably no longer hidden" applies. But the facing rules say this:
A creature can normally target only creatures in its front or side arcs. It can't see into its rear arc. This means an attacker in the creature's rear arc makes attack rolls against it with advantage.
And, in 5E, you don't need to be hidden to sneak attack. You just need to attack with advantage. So, if you are using the facing rules, this might work. The rogue just needs to stay behind the other character. Note, though, that this is very easy to defeat, because:
A creature can also change its facing as a reaction when any other
To me, this supports the basic idea that even with the facing rules, situational awareness is assumed to be high. If someone moves up behind you, you can look at them. And whether or not you agree with that particular interpretation, it definitely means the under-attack player can whip around and look at the approaching rogue. Since it's as a reaction, though, that can only happen once per round, so a really sneaky rogue could possibly draw the reaction by moving in one direction, and then circle around.
To me, this seems kind of rule-bound and weird and good enough for fighting monsters but a little... unsatisfying... when characters are at each other's throats. Make sure your players are both okay with all of this before you set out. (At the very least.)