<b> flawed, see Martin Hairer's comment below.</b>

Step I) Perform substitution $u_t(x) = \exp(\psi_t(x))$. The SPDE for $\psi$ becomes, after dividing by $u_t(x)$,
\begin{equation}
\frac{\partial}{\partial t} \psi_t(x) = \frac {\kappa}{2} \left( \left(\frac{\partial \psi_t(X)}{\partial x}\right)^2 + \frac{\partial^2 \psi_t(x)}{\partial x^2} \right) + (K - \exp(\psi_t(x))) + \sigma \xi(t,x).\end{equation}
This has the added advantage that your noise has become additive, which is usually easier to analyse.

Step II) Note that
$- \psi_t(x) \exp(\psi_t(x)) \leq 0$ for $\psi_t(x) \geq 0$, and $-\psi_t(x) \exp(\psi_t(x)) \leq |\psi_t(x)|$ for $\psi_t(x) \leq 0$. In principle therefore I think you should be able to use an approach similar to the one taken in e.g. X. Mao, <i>Stochastic differential equations & Applications</i>, Theorem 2.3.5, where local Lipschitz property and $x^T f(x) \leq K (1+|x|)^2$ are shown to be sufficient for wellposedness of $d X_t = f(X_t) \ d t + \sigma \ d W_t$. I am afraid I don't know a reference for this result in a Hilbert space context.

Alternatively, you could use Girsanov theorem (Da Prato & Zabczyk, <i>Stochastic equations in infinite dimensions</i>, Chapter 10) to establish existence of solutions without the nonlinear drift part, and then construct an appropriate change of measure.

Hope this helps!