  This paper is concerned with the values of Harish-Chandra characters of a
class of positive-depth, toral, very supercuspidal representations of $p$-adic
symplectic and special orthogonal groups, near the identity element. We declare
two representations equivalent if their characters coincide on a specific
neighbourhood of the identity (which is larger than the neighbourhood on which
Harish-Chandra local character expansion holds). We construct a parameter space
$B$ (that depends on the group and a real number $r>0$) for the set of
equivalence classes of the representations of minimal depth $r$ satisfying some
additional assumptions. This parameter space is essentially a geometric object
defined over $\Q$. Given a non-Archimedean local field $\K$ with sufficiently
large residual characteristic, the part of the character table near the
identity element for $G(\K)$ that comes from our class of representations is
parameterized by the residue-field points of $B$. The character values
themselves can be recovered by specialization from a constructible motivic
exponential function. The values of such functions are algorithmically
computable. It is in this sense that we show that a large part of the character
table of the group $G(\K)$ is computable.
