It's been a while, but I wanted to post an epilogue to this saga as people reading later may be curious how this worked out, what the problem was, etc.
So rest assured, the water cleared up during a week or so of doing the SLAM process described on this site. Everything has returned to normal and we've been enjoying the heck out of the pool this summer.
TL;DR:
I didn't have algae, my cloudiness was caused by a too much clarifier product (Pool Juice 911) since it's enzyme based (enzymes are organics). The SLAM treatment fixed it because the additional chlorine oxidized all the organics away.
Post-mortem w/details:
This all started when I followed the pool store advice to treat for phosphates using their Bioguard Pool Juice Zero product. They recommended I follow that up with their Bioguard Pool Juice 911 product, an enzymatic clarifier according to the label, to clear up the cloudiness brought on by the phosphate treatment. Me, a naïve new pool owner, followed their advice that week and treated the pool for phosphates, waited two days, then treated using the recommended amount of Pool Juice 911 as per their computer's printout instructions.
The following week I took a new sample back to be tested and while the phosphates had dropped considerably (~4k -> ~1.2k), they recommended a second round of phosphate treatment, followed by a second dose of clarifier. (Spoiler alert! This is where I believe the problem stemmed from) I was assured their computer takes into account all the products it tells you to add so that it accounts for say, lowering the pH when you add stabilizer for example.
What I believe happened is that the computer didn't take into account the previous week's treatment using the clarifier. The clarifier has a warning on the bottle that says using too much may cause persistent cloudiness. *facepalm*
I say this because I have never seen any evidence of algae in the pool, no slick surfaces, no growth on anything, nothing that looks even remotely like algae ever showed up, and in the end, we had no dead brown algae to remove.
Thanks again for the great resource and advice from everyone who helped answer my questions and guide me on this journey! Picture attached to show results