may as well A as B / might as well A as B?

應該是：You may call A as B=你可以把A叫做B。（you may ...在這裡是表示：我「同意」或「准許」你去做某事。）
as well = (1) 也; (2)一樣好
He speaks Russian as well.=他也能說俄語。
He can speak Russian as well.=他俄語說得也（與...）一樣好。

Cats and tigers belong to the same family; therefore, you may "as well" call the latter "as" big cats.
=... 你「可以」稱後者為大貓。

...you might call the latter as big cats.=你「可能會」叫後者為大貓。（我認為這事有可能發生）
所以用might與may時的語意不同：可能vs可以。

If you say that you "might as well" do something, or that you "may as well" do it, you mean that you will do it although you do not have a strong desire to do it, and may even feel slightly unwilling to do it.
If I've got to go somewhere, I may as well go to Birmingham.

If you say that something, usually something bad, "might as well" be true or "may as well" be true, you mean that the situation is the same or almost the same as if it were true.
The couple might as well have been strangers.

You use "as well as" when you want to mention another item connected with the subject you are discussing.
She published historical novels, as well as a nonfiction study of women in the British Empire.

see p.1774~5 Collins Cobuild English Dictionary for Advanced Leaners (2001)

（１）ｍay as well A as B
（２）might as well A as B
基本上都有：

既然Ｂ了，不彷（ｍａｙ／ｍｉｇｈｔ）也（ａｓ　ｗｅｌｌ）Ａ吧！
= given B, there is no reason not to A.

的意思．

其中唯一不一樣的是：＂不彷＂與＂不如＂之間的輕重味道．

因為ｍａｙ的說者的肯定性比較強勢，故時常用在偏重＂不如＂的語氣的時候．

例子：
You may as well give me the computer as (give me) the TV set.
既然給我ＴＶ，不如把電腦也給我吧！

你唯一錯誤的地方在於　＂與其做＂Ｂ　的解釋．
這裡沒有　＂不做Ｂ＂的意思．而是＂做Ｂ＂讓＂做Ａ＂更／也有價值．

may / might的句式意思/翻譯都是一樣「做B如同做A」
只是動詞時態和意義上表達: may=可以，might=過去時態的"可以"或者是假設性的"可能"。

因此你原初對這兩字的理解有些誤差了。

由於句字是在說明一件理解的事實或意義，注意前句的時態是現在式=說明一件事實，而不是假設的"命題"或"推想可能"的事，所以後句表示由於前句的定義(貓跟老虎都屬貓科類)，後句的表達是說明實際上"可以"把老虎說成是"大貓"。