Milton,
Please excuse me, but as I see it (YMMV) your math is wrong.
I'm afraid my example is somewhat complicated, because I
pulled the numbers from the thin air and feel sometimes crippled
without a blackboard :)
[BTW: your guess of the minimal set is pretty close, but see section
B below for a smaller set of voters)

::A::
The election rules say that each voter may cast any number 'n'
of votes 0<=n<=3 (the ballot form reads "select at most three").
To further complicate the things, NotA (N) has ambiguous value
i.e. one mark in the N box may count as 1, 2 or 3 votes.
Therefore, there are 15 possible combinations for ticking the
four boxes.
ABC
ABN
ACN
BCN
AB
AC
BC
AN
BN
CN
A
B
C
N
0 (no box is ticked)

The number of Ns counts as a threshold against any candidate. And the
problem stems from here! ABN means an 'explicit' rejection of C, AN
means an explicit rejection of B and C, while AB and A don't mean explicit
rejections. Please notice also a somewhat contradictory effect: a voter
considers A the optimal choice, and B, C not fit for the position. Then
s/he votes AN... but that vote will rise A's threshold.

:B:
By "the most consensual" I tried to mean the candidate with the largest
nonnegative opinions ('for'+'neutral'). In the following example, C has
the lowest number of explicit rejections (all non-listed combinations have
zero votes):
ACN 104
BCN 104
AB 20
A 111
B 97
N 2
Tally:
436 voters.
Threshold: 104+104+2 = 210
A: 235 > 220 → pass (with 106 explicit rejections)
B: 221 > 210 → pass (with 106 explicit rejections)
C: 208 < 210 → fail (with 2 explicit rejections)

:C:
As a conclusion: this curious voting method may take us out of the
impasse now, but has a lot of undersirable properties: is prone to tactical
voting, has no monotonicity, does not satisfy the Condorcet ciriterion, etc.
Once we get through the current election process, discussing an acceptable
voting system seems to be a wise move.

Regards from the Far South,

Enrique