How did 索性 get translated as "you might as well (do it)"
Here’s a highly unsatisfying answer.
《汉语大辞典》has a definition of 索 that defines it as:

索性；干脆。

And they give the example:

《警世通言·庄子休鼓盆成大道》
早知死後無情義, 索把生前恩愛勾。

《现代汉语大词典》also has an entry on 索 that defines it as:

干脆。如：索性；索兴。

Outlier gives the following meaning tree which maps out the development of word meanings and it seems to show that this meaning was originally of a different tone:

MEANINGS
Character meanings
suǒ
1 (orig.) two hands holding a rope made of plant stalks and leaves
2 → rope
3 ⇒ to search, follow a line
4  to ask, demand
suó
1 ⇒ strip, line
2  simply

It would seem that suó was later merged into suǒ.