Red Sea NOPOX is carbon dosing, and quite powerful at that. Don't carbon dose without a skimmer, the skimmer removes the bacteria the carbon dosing feeds. If you carbon dose without a skimmer you may get cloudy water.
Algaefix works very well, but I'm somewhat suspect of its reef safe claims. I haven't used it with live corals but it does seem to do weird things to inverts, making them sluggish. If you try it make sure you get the marine version, the freshwater algaefix WILL knock out your inverts.
Phosguard is effective at tamping down high phosphates, and therefore controlling algae. Make sure you're testing your water to make sure you actually need it. Hair algae can be tenacious once established, even with 0 phosphates it can be a pain to eliminate. Snails and more snails are what have worked for me in the past.
In my opinion lettuce nudibranchs are useless. Sea hares will make an actual dent on hair algae, but there's always the problem of what to do with it when it eats all the algae. Occasionally you'll see one being passed around the forum - search the forum for these threads, not sure if there are any current ones.
Emerald crabs are pretty weak in my opinion as well, just a few hairs short of being a gorilla crab. They can be very effective at tackling algae, and while their skill at removing bubble algae is renowned I'm not sure how well they go after red bubble algae. Like most crabs they're omnivores and border on semi-aggressive.
If I were you I would:
1. Do more research on Marine Algaefix. Perhaps there are people out there who have successfully used it on reefs, though I would suspect not.
2. Invest in a quality skimmer. They can be pricey but make a big difference in water quality, plus it opens up carbon dosing as a method for nitrate/phosphate control.
3. Look into hydrogen peroxide dosing. One of the most popular threads on the subject can be found on this forum. I haven't personally had success with it but many other people have.
4. I know you want to avoid adding livestock, but a few more turbo snail can go a long way towards knocking out hair algae.
