The most efficient way I know to detect whether an element of $K_2(E)$ is torsion is to use the elliptic dilogarithm. This relies however on a conjecture, and it is not an exact method, in the sense that it uses floating-point arithmetic.
Consider the map
\begin{align*}
\beta : F^\times \otimes F^\times & \to \mathbb{Z}[E(\overline{\mathbb{Q}})] \\
g \otimes h & \mapsto \sum_{P,Q \in E(\overline{\mathbb{Q}})} \mathrm{ord}_P(f) \mathrm{ord}_Q(g) [P-Q].
\end{align*}
Bloch has defined a regulator map
\begin{equation*}
\mathrm{reg}_\infty \colon K_2(E) \to \mathbb{R},
\end{equation*}
which can be computed using the elliptic dilogarithm $D_E : E(\mathbb{C}) \to \mathbb{R}$. Namely, given $x \in K_2(E)$, whose image in $K_2^T(E)$ is written as $\sum_i \{g_i,h_i\}$, we have (up to a constant factor)
\begin{equation*}
\mathrm{reg}_\infty(x) = \sum_i D_E(\beta(g_i \otimes h_i))
\end{equation*}
where $D_E$ is extended by linearity by defining $D_E(\sum n_P [P]) = \sum n_P D_E(P)$. The elliptic dilogarithm can be computed very rapidly: writing $E(\mathbb{C}) = \mathbb{C}^\times/q^{\mathbb{Z}}$, one has $D_E([x]) = \sum_{n \in \mathbb{Z}} D(xq^n)$. Here $D$ is the Bloch-Wigner dilogarithm, implemented e.g. in PARI/GP as $\texttt{polylog(2,x,2)}$. Since $|q|<1$, the series for $D_E$ converges exponentially fast.
One should also take into account the bad places of $E$. For each prime $p$, there is a residue map $K_2(E) \to K'_1(\mathcal{E}_p)$, where $\mathcal{E}_p$ is the fiber at $p$ of the minimal regular model of $E$ over $\mathbb{Z}$. It turns out that $V_p := K'_1(\mathcal{E}_p) \otimes \mathbb{Q}$ is nonzero precisely when $E$ has split multiplicative reduction at $p$, in which case $\operatorname{dim}(V_p)=1$. It is also possible to compute the residue map, see Bloch, Grayson, $K_2$ and $L$-functions of elliptic curves - Computer calculations and Rolshausen, Schappacher, On the second $K$-group of an elliptic curve.
Now, it is conjectured that the extended regulator map
\begin{equation*}
\mathrm{reg} \colon K_2(E) \to \mathbb{R} \oplus \bigoplus_p (V_p \otimes \mathbb{R})
\end{equation*}
is an isomorphism after tensoring $K_2(E)$ with $\mathbb{R}$. In particular, and conjecturally, an element $x$ of $K_2(E)$ is torsion precisely when it is in the kernel of the extended regulator map.
If the image of $x$ appears numerically to be 0, then one may try to ascertain that $x$ is torsion in $K_2^T(E)$ by finding Steinberg relations (this becomes a linear algebra problem in the group $F^\times \otimes F^\times$). On the other hand, if the image appears to be nonzero, this can in principle be proved by computing with enough accuracy.
Regarding the torsion in $K_2(E)$ and $K_2^T(E)/ K_2(\mathbb{Q})$, it is known that $K_2(\mathbb{Q})$ embeds in $K_2(E)$ by means of the structural morphism $E \to \operatorname{Spec} \mathbb{Q}$, since this morphism has a section. The group $K_2(\mathbb{Q})$ is infinitely generated (for a description, see Milnor's book Introduction to algebraic $K$-theory). So the torsion of $K_2(E)$ is infinite. I don't know about the torsion of $K_2^T(E)/ K_2(\mathbb{Q})$ in general, but Goncharov and Levin have given a complex computing $K_2^T(E)/K_2(k)$ for an elliptic curve $E/k$, at least when $k$ is algebraically closed, see Theorem 1.5 in Zagier's conjecture on $L(E,2)$. For $k=\overline{\mathbb{Q}}$, this shows (if my computation is correct) that $K_2^T(E)/K_2(\overline{\mathbb{Q}})$ contains a copy of the group $E(\overline{\mathbb{Q}})_{\mathrm{tors}}$.