What to measure??
Commercially available sea salts on the market today vary widely in their chemical composition. Although the principal component is sodium chloride, the remaining portion can range from less than 0.2 to 10% of other salts. These are mostly calcium, potassium, and magnesium salts of chloride and sulfate with substantially lesser amounts of many trace elements found in natural seawater. Though the composition of commercially available salt may vary, the ionic composition of natural saltwater is relatively constant. https://en.wikipedia.org/wiki/Sea_salt
The conclusion from Wikipedia seems to be that the composition of natural seawater is relatively constant (but complex!), while the dry commercial sea salt is quite variable, probably due to differences in production. Commercial table salt is essentially pure for your purposes.
You are "looking to make the perfect flakey sea salt". NaCl tends to crystallize out as cubic crystals, so you are trying to control the crystallization by means of other active agents, which might be at a very tiny concentration. Understanding the chemistry in such depth that you could publish in a respected journal would be possible, but probably not feasible, and maybe not even all that interesting.
Here's what I propose: You have some situations where you take seawater and get some desirable flakey crystals. You will also have some liquid brine left over. Is the active agent all deposited with the solid salt, or is active agent a major component, and retained in the concentrated liquid brine? Test: 1) Recrystallize the solid salt. Does it retain its flakey crystal structure, or have you fractionated the active agent into the retained liquid? 2) Take the liquid brine left over from the first crystallization, add the exact amount of ("pure") table salt that was removed, dissolve with heat, and cool to crystallize again. Do the new crystals have the flakey structure, indicating that the active agent was retained, or do you get simple cubes? The two experiments should give some idea how the active agent behaves. #2 can also be modified by adding more or less pure salt to determine how effective the active agent is in producing flakey crystals.
Hopefully, the active agent will go to one place and not the other, and you will have established some means of control that satisfies your need, For example, if all seawater has the active agent, just at slightly different concentrations, you could take your seawater, precipitate out the first (non-flakey) crop of salt, then add the liquid retained to a fresh batch of seawater and boil it down. In this way, you will have increased the level of impurities. If, on the other hand, the active agent (or lack of an anti-active agent!) has gone into the solid salt, add that to your next batch of seawater to produce flakey crystals.
Another possibility is that the flakeiness is not dependent on the analysis, but on some procedural difference (stirring, rate of temperature decrease, size of batch, seeding, some biologics that come from the seawater that do or do not decompose under your boiling down - seawater is foamy from animals living and peeing in it). In that case, even the best analytical procedures might miss the clue.