I bought that Ballistic Recreations kit a few years ago - used it about half dozen times. A PID is not going to add power to make the Lee Melter work better - it just monitors the temp of the melt and gives power as needed in response to the temp. What I noticed - the B.R. kit came with a thermocouple and readout - so that you would see a number for what exact temp that melt was at. The Lee Melter pot basically has a scale from 1 to 10 - does not actually tell you what temp your melt is at - so you might have the dial on "7.2" to have the melt at 520F. So, if I decided that I wanted the fluid melt at 520 F - would plug all in and eventually get that Lee Melter to get that melted salt to 520 F as per that readout.
But, as I started to do cases, would see the temp drop - maybe down below 500F if you let it go that far - then stop and sit there until the Lee Melter got it back up to that 520 F. In the end, I think I settled on a range - when the readout went below 510F, I would stop - let the thing get back up to 520F - so my "annealing" was actually between 520F and 510F - was not at a precise temperature, as per that readout. I do not think a PID can do a lot better?? It will no doubt apply power faster as the temp starts to drop - but can not supply any more or any extra power than you get from the outlet. A PID might get you "finer" control, but if you are pulling heat out by inserting cold cases, at a greater rate than the Lee Melter can supply - the melt temp is going to drop.
As I understand it, the other part about "annealing" is time at that temp - so I would do case neck "soak" in that melted salt, for count of 5 seconds - is likely much to be argued about "how hot" should things be, and for "how long" to soak the case mouths at that temp.
I am not sure, but I think that "dial" on the Lee Melter pot is like a thermostat for a furnace - is an on-off switch - is NOT a throttle - so when it calls for heat, the thing applies full power to the heating element - until the temp is reached - then the element is shut off - is On-Off - not partially on - I think the various PID can vary the amount that goes in - to get "finer control" - but can not make any more power than fully "On". Would be like thinking to turn your wall thermostat for furnace way high - to get house to warm up faster - does not work that way - furnace is On or Off - is no "throttle" to make it put out more heat the higher the thermostat is set.
I would suggest that your true concern is to know what is the temp of the melt. The Lee Melter pot does not tell you that. So, I suspect cheapest might actually be like a thermometer that reads in that sort of temp range of the salt bath - like the old school Lyman dial readout thermometer for lead casting - I think that would read up to 1000F, but I actually do not know how "accurate" they were, although "modern days" - is likely a thermocouple readout, like on a PID, might end up to be cheaper to buy - but I really do not know if they are any more "accurate" or not - they do give a discrete number though.