In the "classical theory of topological vector spaces" the questions like this are intricated (in my opinion, this is an artifical complexity, the Nature can't be so complicated). But in the [theory of stereotype spaces][1] they become simple: for each stereotype space $X$ (including ${\mathcal S}({\mathbb R}^k)$, since it is also stereotype as a Fréchet space) the pairing
$$
(x,f)\in X\times X^\star\mapsto f(x)\in{\mathbb C}
$$ 
is a [continuous bilinear map in the stereotype sense][2], and, as a corollary, it can be extended to a continuous linear functional on the ["projective stereotype tensor product"][3] $\circledast$ (this is an analog of $\hat{\otimes}_\pi$ in the stereotype theory)
$$
\operatorname{cont}: X\circledast X^\star\to {\mathbb C}.
$$
This functional is called a "contraction", you can look at the details in [my paper of 2003][4] (page 265).

If you want to define a trace for all operators $\varphi:X\to X$ that are images of the tensors $\alpha\in X\circledast X^\star$ under the [Grothendieck transformation][5] $X\circledast X^\star\to{\mathcal L}(X)$, then your space $X$ must have the [stereotype approximation property][6]. As far as I know, nobody was interested up to now, whether the space ${\mathcal S}({\mathbb R}^k)$ has the stereotype approximation, but at first glance this is true: one can try to use the same trick as I did in [my paper of 2018][7] for the space ${\mathcal C}(G)$ of continuous functions on a locally compact group $G$. Also Albrecht Pietsch writes in his [Nuclear locally convex spaces][8], 10.3.2, that the [Hermite polynomials][9] form a basis in the space ${\mathcal S}'({\mathbb R})$ (which coincides with the stereotype dual space ${\mathcal S}({\mathbb R})^\star$). I think something similar must be true for each ${\mathcal S}'({\mathbb R}^k)$ (with arbitrary $k\in{\mathbb N}$), and if so, then ${\mathcal S}'({\mathbb R}^k)={\mathcal S}({\mathbb R}^k)^\star$ and ${\mathcal S}({\mathbb R}^k)$ have the stereotype approximation property.


  [1]: https://en.wikipedia.org/wiki/Stereotype_space
  [2]: https://en.wikipedia.org/wiki/Stereotype_space#Universality_of_%7F'%22%60UNIQ--postMath-0000014E-QINU%60%22'%7F
  [3]: https://en.wikipedia.org/wiki/Stereotype_space#Ste_as_a_*-autonomous_category
  [4]: https://link.springer.com/article/10.1023%2FA%3A1020929201133
  [5]: https://en.wikipedia.org/wiki/Stereotype_space#Grothendieck_transformation
  [6]: https://en.wikipedia.org/wiki/Stereotype_space#Stereotype_approximation_property
  [7]: https://arxiv.org/abs/1803.01340
  [8]: https://www.springer.com/gp/book/9783642876677
  [9]: https://en.wikipedia.org/wiki/Hermite_polynomials