I think this deserves further discussion.
You say that HU with a hand like that our main concern is to get to showdown so we chould check. Then you say that, depending on the opponent, we should check raise the turn?
Seems very inconsistent to me. Are we calling a 3-bet? If not then seeing showdown is apparently not imperative. Note that we can see a showdown for those same 2 bets we invest on the turn by CRing. If we are calling down a 3 bet then we're prepared to pay 4 bets to see showdown instead of 2 and are likely just about dead. Not a situation I'd want to be in.Ok. I should clear this up a bit. Your main goal should be to both extract bets and get to showdown without spewing. It's a tough balancing act.
Anyway, Once we get to the river, I think that the only lines we consider should be ones that get us to showdown. A check/raise/fold may be ok too, since a raise after we bet given the action so far means a lot less than a raise after we check/raise. Even though we have better odds if we chekc/raise and get 3-bet, the information we have is much greater too.
Same line of reasoning goes for the turn. You check/raise certain players that bet if checked to. This makes money the times they have nothign, and extracts value when they have a weaker made hand or draw(most of the time). And if we get 3-bet on teh turn, it is player dependent, but most players aren't 3-betting the turn at this level without the goods, so we can safely fold. Of course some players just go apeshit HU, in which case it is probably best to just keep betting and let him raise. If he does, simply call.
I see the incosistency in teh arguement. The point I'm trying to make is that HU situations are very player dependent. Also, most players will make plays by raising you wehn you use the raise-bet-bet-bet line, but won't even consider making a play if you check/reaise them on a big street.
If we know that our opponent is at all rational, and this is a very liberal definition which probably encompasses about 95% of the online world, then he simply cannot raise that river with a hand we beat. If he pulls the bluff of his life then good for him, all good players sometimes fold the best hand. But he's not running a move there nearly often enough for us to call the raise IMO and I'm not particularly worried about him making one. I think this is completely wrong. I will often raise players on the river with a busted draw if I've seen them bet/fold. I will raise an even wider range of players on the river with any hand I plan to call with anyway after this action. That means if I have somethign like 44 here, and I plan to call the river, I'm raising against just about everyone who isn't a showdown monkey. I think your definition of rational is off. Sicne so many players bet/fold way too much, raising the river is actually quite a rational play.
I know what you're going to say, most people I play against don't do this. But how do you know? You are bet/folding so much, you never can tell. I've seen even seemingly passive players take this line from time to time. It doesn't mean you should call here vs a very passive player (you shouldn't), but against most players in teh online world, you have to call this river if you get raised.
Now that doesn't mean you have to call all rivers with anything marginally showdownable when you get raised - you shouldn't. Sometiems the action is such where you know your hand isn't good enough. But in HU blind battles like this, where all you've done is bet-bet-bet, you really can't fold to a raise against an unknown player.
If we assume that BB will call the river with any pair, but will check behind with a pair of 7's or worse and will only raise us with hands beating us then bet/fold is unquestionably the best play. 
Who would question that?!
I think that's a pretty reasonable assumtion to make even though we can obviously never be 100% certain how a player will act.Obviously I think the above is a terrible assumption.
I've also already covered the few situations where checking is better, such as when our oppnent holds 88/99/TT and checks behind or holds a draw such as 89 and would have fired a final bullet but folds to our lead. However, I think that on balance those very rare exceptions don't make up for all the times we let a hand we beat off the hook for a bet and we're losing another bet to a better hand than us nearly 100% of the time whether we check or bet anyway.I think you also underestimate the times your opponent has complete air here. Obvoiusly checking is better then. Also, a lot of players won't always check behind weaker made hands, they will bet for value sometimes too.
FWIW I don't even think we have the best hand here half of the time when we are called and its still a bet! (see pg. 211 of the "Theory of Poker" for a thorough discussion of betting on the end as an underdog) It basically boils down to the fact that your opponent will bet with less hands than he'll call with and you are forced to call the bet anyway.I understand fully the concept of betting OOP, when you're an underdog. But in this instance, I think you are completely wrong if you think that we are an underdog on this river.
BTW, I think my earlier assessment of check/call was off teh mark. I still think it is better then bet/fold, but I think bet/call is the best option. Without a read on how aggressive your opponetn is, it is very hard, if not impossible to determine whether checkingn or betting is best (it depends on how many hands he'll bet if you check). In either case, I thinkn folding this river no matter what line you take vs an unknown is a mistake.
Also, I think that your advice is 100% correct if you are up against an utter calling station. Obviously, checking here against a calling station is a very bad play. Betting is the only option. And folding to a raise is smart as well. Unfortunatley, we don't know we are up agianst a calling station. WE could be playing against a weird aggressive player.