First you started with no CYA and using bleach...
With no CYA (and with 2000 ppm salt), you would outgas virtually all of your FC in about 10 hours! Even worse, without CYA you can lose half of your FC in just 30 minutes due to sunlight! So even though you covered your tub when not in use, even short-term use during the day would really eat up the FC very, very quickly if there was no CYA.

Then you used powdered stabilized shock...
Powdered stabilized shock was probably Di-Chlor (since that doesn't change pH much and it dissolves quickly). The small size of your hot tub means that adding only 1/4 ounce of Di-Chlor would increase your Free Chlorine (FC) by 8.6 ppm while also increasing your Cyanuric Acid (CYA) by 7.9 ppm. So basically when using Di-Chlor, you add about as much CYA as you do FC, but the FC goes away while the CYA does not!

So it's no wonder that the chlorine wasn't effective after a few doses of Di-Chlor. Your CYA levels probably are through the roof -- if you can test that and let us know, that would be very interesting. As for BleachCalc, my guess is that your hot tub is larger than 120 gallons -- are you sure of that number as this is small and would be for a 2-person hot tub (5-6 person tub is around 375-400 gallons)?

A high CYA level would explain why adding chlorine wasn't effective at killing hot tub itch. That still leaves the issue of the chlorine getting used up so quickly even after you started using Di-Chlor. The high salt level would certainly contribute to chlorine getting outgassed, but the high CYA would cut that down significantly and would also protect the chlorine from sunlight. So that leaves the other source of chlorine consumption, namely killing bugs and oxidizing organics. I'll bet that your CC levels are high and that the chlorine is struggling to keep up with everything (such as sweat) introduced into the relatively small volume of water.

When you were using Baquacil...
I don't think it's so much that Baquacil was better at killing Pseudomonas aeruginosa than chlorine, but that when the chlorine was used up (and not there) or when it was in the presence of a lot of CYA, then it was not effective. So the key is to get the chlorine level up and to stay up and to not have too much CYA present. Also, when you were using Baquacil, did you add both the biquinide (PHMB) and hydrogen peroxide? The former kills bacteria and algae, but only the latter oxidizes organics and most of the junk in spa water is organics (sweat).

My guess at this point is the following. When you used Baquacil, you were adding biguinide (PHMB) but not hydrogen peroxide. So while you were killing bacteria (including hot tub itch) and preventing algae, you weren't oxidizing organics (e.g. sweat) so when these built up that caused the water to get cloudy. When you switched to using chlorine, most of it is getting used up oxidizing the organics and perhaps you never got to a point where you oxidized all of them so just saw perpetual chlorine demand (and with no CYA initially and high salt levels you were also using up chlorine through outgassing as well as breakdown from sunlight). After you added Di-Chlor, the CYA made the chlorine ineffective and you got hot tub itch.

Richard