There are three kits that I am aware of (availability aside). In addition to Red Sea and Salifert, there is also an Elos phosphate test kit, Of the three, I've only ever used the Red Sea. Based on the scientific literature, and an awful lot of conjecture on my part, I'm pretty sure that they all three use the same fundamental analytical technique for potassium (precipitation of the sparingly soluble potassium tetraphenlborate salt). Elos then basically uses turbidimetry by measuring Secchi depth. Red Sea takes the approach of adding an excess of sodium tetraphenylborate, and then back-titrates the remainder left over after the potassium precipitates, using an indicator dye and a chemical very much like Lysol as the titrant, but because the titrant reacts with the precipitant, that tedious filtration step is necessary. Salifert seems to get away without having to do the filtration step by using another chemical that blocks the reaction between the titrant and the precipitant (that's what I hope to confirm tonight). The Red Sea and Salifert kits probably both use the same indicator dye, but the Salifert approach adjusts the pH to a range where the color change is much more distinct and easier to detect (yellow to blue rather than bluish-purple to blue).
If I'm right in my conjecture, then it should be possible to make the Red Sea kit behave like the Salifert kit (no filtration necessary and better color change) with the addition of a couple of the reagents I just brought home tonight.