Finding out the composition of seawater/brine?
Your budget should allow you to measure  $\ce{Mg^{2+}, Ca^{2+}, Li+ and Cl-}$ in-house using ion specific electrode analysis. There will certainly be some training and maintenance involved, but anyone with a bit of a chemistry background will be able to learn this technique. Adding to the simplicity, and thus the ease of training, is the fact that your samples are all of a fairly similar composition.  

Ion chromatography, which would basically require one set up for anions and one for cations, would break your bank in a hurry unless you have an experienced IC analyst with some time to spare and access to good used equipment.  Frankly, this just doesn't seem like a realistic approach.  

One thing to keep in mind is that your budget will buy a lot of analysis by people with expertise in whatever analysis you choose to perform. They will even help you choose the best analytical technique for your needs.  

My advice is to look at outsourcing first, particularly if you are going to be analyzing fewer than 10-20 samples per day.  If you will be analyzing a large number of samples, or if you need very quick turnaround times, then I suggest looking into ion specific electrode analysis.