The best/most accurate of the phosphate test kits is Sera brand. (unless you get a digital meter which is much more expensive) This is also a pretty simple test to perform. I'm not aware of any liquid test that measures to the decimal in regards to phosphate, but that is usually not needed, either. If the test is read under a fluorescent light and set on a white piece of paper (such as computer printer paper), any trace of blue in the results will indicate some level of phosphate in the water, which is all that is needed.
IF phosphate is the culprit or one of the contributing factors there are a number of very good filter medias for removing it easily. If your water tests positive for phosphate I can then suggest those to you.
While addition of tap water seems to increase the problem that doesn't necessarily mean that the tap water itself is actually the origin of the problem. It could be as simple as the clean water having more oxygen in it that contributes to the renewed algae bloom. I would suggest testing both tank and tap water for phosphate, but test immediately after removing the water from the source. The longer water is let to sit out the more altered the test results will be. Water chemistry is ever changing.
I am concerned about the mention of the brown debris coming from the filter media. I am wondering if this is algae, algae die off, or some other type of organic matter, and at present there isn't enough information to determine that accurately.
In regards to the live plants... 3 live plants in a tank aren't going to do much in way of removing nutrients from the water. It takes a lot of plants to utilize a small amount of nutrients.
I'm going to suggest we wait for those phosphate results before taking any further steps in correcting this problem because we want to find the source and eliminate it rather than treat it only to have it start again. I also suggest doing smaller and more frequent water changes in the mean time... instead of 15%/wk try 5 - 10% each day. This will lessen the amount of change in the water each time and will also help to add oxygen for the fish more regularly which will help to sustain them until we get to the root of the problem. If you have the ability to add an air stone I will also suggest doing that, but with an air control valve so you can regulate the rate of bubbles being produced. You want slower bubbles instead of a faster stream. The faster they rise to the top the less oxygen they diffuse into the water. Fast bubbles will increase the rate of gas exchange at the surface but will do little to nothing for adding oxygen into the water column. Faster bubbles may also further stress the fish.
I will check back here tomorrow.