[Mook]

There's no need to overthink the strategy. You just need to consider the differences from a standard limit game and then figure out how best to adjust to those differences.

And really, there are only two differences from a traditional $4-$8:
(1) The bet preflop is $2 instead of $4.
(2) The turn bet is $4 instead of $8.

Regarding point (1), you should be seeing the flop with more speculative hands than you normally would, not fewer. The EV of these hands is most heavily influenced by what you have to invest to see the flop (which is why they're very profitable in passive games and generally unprofitable in aggressive ones). On average, you'll have to put in only half as much, so any hand that returns slightly over half as much from the pot on average as it would in a normal $4-$8 game is playable. ("Slightly over" because the smaller turn bet means you won't extract quite as much value from them when you hit.) I wouldn't be surprised if this made some complete suited garbage like 53s profitable, certainly from LP. Definitely I'd think any pocket pair would be playable from any position as long as the pot hasn't been raised, and probably even against a raise, since you'll probably still have your necessary 5- or 6-way action.

Also, you have to be careful about raising PF with hands that need to play defense after the flop. You won't push too many people off a hand for an extra $2, and all you'll wind up doing by bloating the pot is giving weak drawing hands even better odds to chase. I'd really have to wonder even whether AKo is worth raising preflop in this limit structure. But hands like QJs, JTs - yeah, blast away, since your "implied equity" in the final pot is probably well above average preflop with these holdings.

Point (2) again favors drawing hands over made hands - primarily because many drawing hands that wouldn't have odds to chase to the river for $8 will have odds to do so for $4. Backdoor draws, especially, will add significantly more value to a hand than they do at standard limit, because when you get halfway there on the turn it will then cost only half as much to see the river, yet when you hit you'll get your opponent(s) for the full $8. Hands like third pair plus backdoor straight draw or gutshot plus backdoor flush draw might be worth chasing to the turn here even when the pot is a couple bets short of offering you "proper odds" based on a "standard" accounting of backdoor draws.

That's what I can come up with off the top of my head - there may be some other more minor adjustments.

Mook