One of the answers at Which essential amino acids are the most likely to be deficient in a vegan diet? suggests beans are the best source of essential amino acids requirements in vegan diets.
Is there a bean or combination of beans that would be the best from a nutritional view in a vegan diet?
The best bean is variety
The main concerns with bean are that many are incomplete proteins: they are relatively low in Lysine and Methionine.
However, there are some exceptions. Most notably Soy is a complete protein with high levels of Methionine so Soymilk, Tofu, Edamame, and Natto are all good protein sources. Similarly, Chickpeas (garbanzo) are higher in Lysine than most beans, while not complete for essential amino acids there easier to complement. Not strictly beans but Quinoa, Buckwheat, and Quorn are all complete protein sources too.
Another consideration is that Methionine and Lysine are plentiful in other foods and easy to complement. Eating beans with Rice, Wheat, or Couscous would get you your full set of amino acids in one meal. Note that Corn is also Lysine deficient which is a problem in some diets.
The best approach to complete proteins is variety, then you ensure you get a variety of other nutrients (vitamins, minerals), and get to eat interesting things instead of the same thing everyday. Soy has become the de facto vegetarian protein substitute for good reasons but a health diet is a balanced one and there's no reason not to give other beans a try.
The soy bean provides more protein (and lysine) than other beans.
On a comparison of protein per gram, soy beans excel with 36 grams protein (of which 2.7 grams lysine) per 100 grams of beans. The next runner-up is mung beans which provide 24 grams protein (of which 1.7 grams lysine) per 100 grams beans. By this comparison soy beans have 50% more protein than any other bean.
Or when comparing protein per calorie, soy beans still win with 8.2 grams protein (of which 0.6 grams lysine) per 100 kcal. The next runner-up is mung beans again with 6.9 grams protein (of which 0.5 grams lysine) per 100 kcal. It's impressive that soy beans still win even though they contain more fat than most beans.
Soy beans provide more of the essential amino acids because the total protein content is so high.
Okay, but other than soy?
Mung beans (aka green gram) and kidney beans are pretty much tied for second place with very slight differences in the amino acid profiles between them.
All beans are deficient in methionine, compared to other essential amino acids they contain and their fulfillment of recommended daily intake (RDI). Check out these charts based on a recent scientific assessment.
Also, you may notice lysine deficiency is only relative to other other essential amino acids, which are fairly abundant in beans. One cup of cooked beans (most types -kidney, pinto, navy, black- and also including lentils, soy and chickpeas) provide 45-63% of lysine RDI, but only 21-37% RDI of methionine. The results may vary if you consider different RDIs for the same amino acid; take methionine - ranging from 13 mg/kg to 19 mg/kg - the previous calculations seem based on the lower RDI.
So, to answer your question, no, there is no single bean or combination of beans that provides optimum intake equilibrium for essential amino acids. Unless you eat 3-4 cups of cooked beans everyday (a lot), you won't get enough methionine. There is less (on no) concern for lysine or even the rest of essential amino acids. Best to pair beans (any type) with good sources of methionine. Soy is best bean choice for methionine, but you should still pair with brazil nuts for instance.
