Some thoughts based on what I understand meaning it might be all wrong, you'll see ... (and this is one of those questions where, without doing one's own full exploration and analysis, it's hard to make firm recommendations).
- you have 4206 observations but also 861 speakers so there aren't many observations per speaker (on average at least);
- from how I understand this (recall caveat from above), your main predictors of interest are the LexDiv variables but actually also SemYear (?). Aren't you trying to find out how the writing proficiency goes up (as measured by the LexDivs) and over time (as measured by SemYear)? If the answer to that question is yes, then
-- wouldn't SemYear need to be a fixef predictor, not a ranef?
-- wouldn't you need random slopes for all 4 predictors of interest (to rule out anticonservative effects)?
Generally, tho, I don't see how you can ignore student as some kind of ranef source (which is what your main question was, so I think you need some version of option 1) and, you didn't ask about that but I'll say it anyway because it might explain the low R^2^~marg~, the relation between your LexDiv predictors and FairAverage may be stronger than you so far think, but curvilinear.
Just my $0.02, like I said, very hard to say in the abstract ...