I don't know about shoving the flop, but that's exactly why I'd play it aggressively and wouldn't let it draw to the river. By the river, the only hand he's likely to call me with is one that beats my two pair.
That, or I'm willing to give it up should the board look scary enough for my hand. If I flopped two pair with drawing possibilities for my opponent, I'd rather take the pot than let my opponent draw out on me.
On a board where two pair is very likely to hold up, I'd slow play.
You contraddict yourself a couple times throughout your 3 posts, but that's not important. What I find really wrong is this irrational fear of getting outdrawn, to the point that you're willing to give up on value. I tried to explain in my previous post that this is not a winning strategy when playing HU.
Flopping 2 pair in a HU match, whether it's top 2 or bottom 2, is a very strong hand. You can get outdrawn occasionally (yes, your opponent can make a straight or a flush) but the percentages of that not happening are always in your favor. Just because the board has 2 cards of the same suit on the flop doesn't mean your opponent is on a flush draw. There is only a 5% chance of that. The biggest danger, instead, when flopping bottom 2, is that the board pairs and you get counterfeited. Again though,
odds of that happening aren't exactly huge.
You seem to think that aggressively is the best way to play this, because you don't want him drawing to the river, so you want to take this down immediately. That's saying you're happy to win the 120 in the pot, after flopping 2 pair. This, IMO, is a huge mistake. You're not going to hit many hands, and getting value from those you do hit is essential. The way to obtain that (value) is subordinated to how you've been playing so far, and how he's been playing. If you've been aggressive on every flop, then go ahead and bet this one too. But if you have checked the majority of flops, and bet only when you hit, you shouldn't make an aggressive bet here after his check just to get the 120, unless you've seen him check the flop previously when he actually had a hand.
As I previously said, HU is a different animal. It's a mix of mind games and other strategies that are all put forward with the intent of getting the most value you can when you hit, and stealing pots when you miss. If you become predictable with your betting patterns then you're depending excessively on the cards to win. It's very important to keep him guessing and make yourself hard to read. At times you need to decieve your opponent in order to get chips. In the OP this villain was described as aggressive, shoving into many pots. You have a hand where you need him to get in his comfort zone and induce him to do just that, either a shove or an overbet. The risk of getting outdrawn isn't huge and on the flop you have to take it if you want to make the most. This huge fear of getting outdrawn won't take you far in HU matches, and it's certainly the wrong strategy against an aggressive idiot like this one.
This is not a full ring game, it's HU. Like it or not, HU you have to slowplay at times in order to achieve chips. Heck, people slowplay top pair or overpairs in HU, leave alone 2 pair. I do agree though that on the turn he should have raised, for reasons I already mentioned before. At some point you need to start betting, although I have to admit that if he got villain to shove with a lesser hand (and I believe he did) by playing it this passively, his turn call was genius.