What is the origin of the phrase "to see it through"? How early was it invented? Would it sound out of place in an attempt to emulate older (200–400 years older) English?
Yes, the words "see it through" are very rare before 1800 in any meaning:
There may have been an earlier way of expressing this with different words.
The Ngram doesn't discriminate between meanings, and includes lots of results for "see it through [some medium]", such as "... see it through thy tears", "... they cannot see it through your spectacles", "... see it through that aperture", "for we see it through a glass darkly".
The concept will be much earlier, but with this exact "see it through" wording, the oldest I found for the meaning "to continue with something until the end" is in a possible use from 1726's A Further Answer, Being a True Representation of Mr. Worger's Case by Dr. John Gray:
From the 1776 Parliamentary Register; Or, History Of The Proceedings And Debates Of The House Of Commons:
This is interesting. It discusses a man determined not to give up a dispute, but "he was bound to see it through" the medium of war. There are several other "see it through the [constitutional/proper/false/war] medium" around this time, but this one mixes determination to complete with the medium he must use to finish it. But still, it's "seeing through a medium".
Here's another interesting find from 1833's The Miscellaneous Works of the Rev. Matthew Henry, which may or may not show the root:
The first clear use is in an 1836 letter in The Monthly magazine, or, British register, Volume 22: