The way I understand it, lithium requires blood testing because the therapeutic dose is close to the toxic dose. Is it monthly testing? Maybe at first while the optimum dose is being found. I think some blood tests may be needed with Depakote because it's hard on the liver. It's been a long time since I've taken it. If I remember right, my doctor tested my liver enzymes before putting me on it to make sure I was in good order before starting it.
Couldn't say which is best to use but Lamictal was the most effective for me as well as other people and generally used more than the others. Trileptal unlike Tegretol (which its clinically similar to) does not require a blood test. Neurontin and Topomax don't but they are generally used as adjuncts. I was not able to tolerate Keppra because of personality changes and its not used often for this exact reason. The mood stabilizer I take now Clonidine is rarely used because it can cause depression and its a blood pressure medication in its original use with the potential of some severe side effects. I don't believe the medication Lovaza requires blood tests but like the rest of the mood stabilizers that would be listed its use as a mood stabilizer is experimental. However with many of these medications, Lamictal included, a blood test before starting the medication as well as at yearly intervals for whatever a doctor would determine but especially liver function is usually recommended. You could ask your psychiatrist about all this.