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}{\partial x}\right)^2 + \frac{\partial^2}{\partial x^2} \right) \psi_t(x) + (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, Stochastic differential equations & Applications, 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, Stochastic equations in infinite dimensions, Chapter 10) to establish existence of solutions without the nonlinear drift part, and then construct an appropriate change of measure.

Hope this helps!

flawed, see Martin Hairer's comment below.

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, Stochastic differential equations & Applications, 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, Stochastic equations in infinite dimensions, Chapter 10) to establish existence of solutions without the nonlinear drift part, and then construct an appropriate change of measure.

Hope this helps!