Does the pad look similar to this?
If so, ditch the stock sponge and just run the filter pad. Use it going forward, just cut it to size and replace it every 2 days or so. You may need to do it every day until you get the water cleared up. This stuff is not meant to be reusable and it's cheap for a reason, so you can just throw it out once it's done. Sounds like your LFS is just trying to get your money by telling you to replace it so quickly, so stay away from them.
Have you gone lights out yet for an extended period? If not, I think it's time. Try turning your lights off for about 72 hours. By lights out, I mean no light at all! Pull the drapes in the house if necessary! This isn't going to hurt your corals at all, they can go even longer if need be, but start with 72 hours. Once complete, due about a 10 gallon water change. The corals you have listed are hardy and should be fine with that amount.
No need to feed the tank right now, as it sounds like you have plenty of food for what you have in there given the current situation. This is going to take a little bit of time to get rid of, so when you turn your lights back on, go with 4-5 hours to start with (I'm leaning towards 4). Let the tank adjust for a week or so. Then make the decision to up it from there in minimal increments.
Algae needs a nutrient source (nitrate and phosphate) and a light source to survive. It has it all right now. The reason you have no nitrate or phosphate reading is due to the algae literally taking it all in. In larger reef tanks, many aquarists employ a refugium, where water from the display tank passes through a section containing some sort of macro algae to absorb nutrients, thus keeping their display tanks clear of nuisance algae (in theory)
All that being said, this is just my opinion. This is an advice forum, and it is just that, my advice. Hopefully it works, should you choose to try it. I'm sure you will work regardless and I'm looking forward to seeing your tank as it matures!! I love my Nano Cube and I love getting ideas from others!! Consider me subscribed!