I just finished teaching a course using our local lake (Lake Winnebago) water - it's not seawater, but hard as anything!
I used the Hach and LaMotte reagents with my spectrophotometers (Genesys from Fisher). My students had to find the correct chemical reactions for their lab reports.
My original procedures came from the Standard Methods for the Analysis of Water and Wastewater book (I think mine is the 17th edition).
What we did:
Potassium by precipitation using sodium tetraphenylborate with absorbance at 750nm taking the place of turbidity.
Chloride (low levels in the lake) by precipitation with silver nitrate, filter, dry, and weigh (Gravimetric analysis).
Alkalinity by pH titration (graphing pH vs. volume acid added, taking first derivative to find inflection point - tedious but great data!).
Phosphate by absorbance using the molybdate reaction (Hach reagent packs).
Calcium and Magnesium by EDTA titrations (one at pH=10 for both Ca+Mg, one at pH=12 for Ca only). We also did Ca and Mg by AA.
Nitrogen cycle compounds (ammonia, nitrite, nitrate) by absorbance and ion selective electrode (NH4 and NO3 only) methods - I made the nitrite reagent (Greiss) but used the Hach reagent packs for the other two.
I'm also setting up some of these for a local club meet in January - so I will soon have procedures written for seawater - pm if you want a copy when I have them.
Kevin