The fact that the character is a Quaker is very relevant. Quakers used thee for both the nominative and objective cases long after most other English speakers had stopped using thee and thou and started using you. See this blog entry,
So using the archaic 2nd-person pronouns was not a mistake. What may be a mistake is the use of thee instead of thou for the nominative — this shift of Quakers to using thee rather than thou took place approximately one hundred years after the novel was set, in the late eighteenth century.
On the other hand, the author may have deliberately decided to use this anachronistic language for artistic reasons. Quaker speech would definitely have stood out at the time, because Quakers used thou and thee for everybody, even on formal occasions. However, if the author had used this more authentic mode of Quaker speech, modern readers might not have realized that the Quaker characters were speaking any differently from the other characters, who would have used both you and thou, depending on the occasion and how well they knew the person they were taking to.
From the blog:
Probably no word is more closely associated with Quakers in the popular imagination than thee, although very few Quakers actually still use it. The fact that Quakers continued to use thee long after most of the rest of the English-speaking world abandoned it is somewhat remarkable, but what I want to focus on more particularly here is the fact that even though Quakers retained thee, they eventually dropped thou (at least in colloquial style), using thee in its place. A related change is the loss of the traditional -est second person singular verb ending, and its replacement with what looks like the third person singular form. The result is that in Quaker speech, one says Thee goes to meeting instead of Thou goest to meeting.
The blog goes on to say:
It should be recognized that nominative thee is not exclusive to Quakers, but is used in a number of regional English dialects, especially in western England.
and
the use of nominative thee became popular among Quakers in the late 18th and early 19th centuries.
