Absorbing boundary conditions have been extensively studied for diffusion equations (e.g., Solving the diffusion equation with an absorbing boundary - but you can find much more by googling), of which Schrödinger equation is a particular case (although with a complex coefficient in front of the time derivative.) So mathematically/computationally this should not present much difficulty.
The difference between reflecting and absorbing boundary conditions is that in the former case we set the current across the boundary to be zero, whereas in the latter case the density at the boundary is zero. Note that, if we solve with absorbing boundary conditions, we are dealing with either a relaxation problem - where after some time the particle density will be zero everywhere - or with a (steady) flow - in which case we need also a source of particles. The latter case seems pertinent to modeling a two-slit experiment - particles injected by something like a point source on one side of the slits, and absorbed by a boundary (e.g., photographic plate) on the other side.
In fact, such steady flows are routinely analyzed in various studies of quantum transport, i.e., electron flow in devices, notably semiconductor devices. So this might be another direction to search for reading materials, although often such studies are done not in terms of wave functions - e.g., a source and a sink of electrons can be modeled as different chemical potentials on the two sides of the device, whose difference is equal to the bias voltage (discussions can be found in any book on semiconductor devices - starting with Shockley diode equation, pn-junction, etc.)
Update
As discussed in the comments, such problems are routinely modeled in Fock space (although rarely in position representation for the system of interest.) E.g., one could write a following second quantized Hamiltonian:
$$
\mathcal{H}=\int dx \Psi(x)^\dagger h(x)\Psi(x)+\sum_k\epsilon_k a_k^\dagger a_k + \sum_k \left[v_k^*a_k^\dagger\Psi(0)+v_k\Psi(0)^\dagger a_k\right]
$$
where $\Psi(x)^\dagger,\Psi(x)$ are the creation and annihilation field operators for the system of interest, whereas $a_k^\dagger,a_k$ describe the states of the photodetector, coupled to the system at point $x=0$. The states of the photodetector would be typically assumed to be very dense and quickly relaxing to equilibrium, so that, once the particle is absorbed, it never experiences a "revival". Such an ideology of a "bath" is commonly used to describe atomic emission lines, or level broadening in semiconductors systems (see, e.g., the approach by Gurvitz here and here or the Green's function approaches, see also the derivation in this answer.)
The particularity in our case is that we do not want to write the equations for the averages, correlation/Green's functions, or density matrix, but for the wave function.
In Heisenberg picture the equations for the field operators resulting from the above Hamiltonian are:
$$
i\hbar\partial_t\Psi(x,t)=h(x)\Psi(x,t)+\delta(x)\sum_kv_k a_k(t),\\
i\hbar \partial_t a_k(t)=\epsilon_k a_k(t) +v_k^*\Psi(0,t)
$$
Formally solving the last equation we have:
$$
a_k(t)=a_k(0)e^{-\frac{i\epsilon_k t}{\hbar}}-
\frac{iv_k^*}{\hbar}\int_0^td\tau\Psi(0, \tau)
e^{-\frac{i\epsilon_k (t-\tau)}{\hbar}},
$$
and the resulting equation for $\Psi$ is
$$
i\hbar\partial_t\Psi(x,t)=h(x)\Psi(x,t)+
\delta(x)\sum_kv_k \left[a_k(0)e^{-\frac{i\epsilon_k t}{\hbar}}-
\frac{iv_k^*}{\hbar}\int_0^td\tau\Psi(0, \tau)
e^{-\frac{i\epsilon_k (t-\tau)}{\hbar}}
\right].
$$
We chose as an initial state a state containing only one particle, not yet absorbed by a photodetector:
$$
|\phi(x)\rangle=\int dx\phi(x)\Psi(x,0)^\dagger|0\rangle,
$$
where $\phi(x)$ is a scalar (wave)function, whereas $|0\rangle$ is a vacuum state with no particles:
$$
\Psi(x,0)|0\rangle=a_k(0)|0\rangle=0.
$$
We thus obtain the Following "Schrödinger" equation:
$$
i\hbar\partial_t\phi(x,t)=h(x)\phi(x,t)
-i\delta(x)\sum_k\frac{|v_k|^2}{\hbar}\int_0^t d\tau\phi(0, \tau)
e^{-\frac{i\epsilon_k (t-\tau)}{\hbar}}.
$$
We now performs some rather aggressive by typical approximations to assure the irreversibility of absorption and simplicity of treatment, taking
$$
\sum_k\frac{|v_k|^2}{\hbar}e^{-\frac{i\epsilon_k (t-\tau)}{\hbar}}\longrightarrow
\frac{\rho|v|^2}{\hbar}\int\frac{d\epsilon}{2\pi}e^{-\frac{i\epsilon (t-\tau)}{\hbar}}=
\frac{\rho|v|^2}{\hbar}\delta(t-\tau)
$$
($\rho$ is the density of the states of photodetector, assumed to be constant, along with the matrix element $v$)
The Schrödinger equation now takes form
$$
i\hbar\partial_t\phi(x,t)=h(x)\phi(x,t)
-i\Gamma\delta(x)\phi(0, t)=\left[h(x)-i\Gamma\delta(x)\right]\phi(x,t)=
\left[-\frac{\hbar^2\partial_x^2}{2m}+V(x)-i\Gamma\delta(x)\right]\phi(x,t).
$$
Integrating the equation over a small region around $x=0$ we can obtain the boundary condition (for more details see this answer about scattering from Dirac delta-potential.)
It is worth noting that we have arrived to a Schrödinger equation with a non-Hermitian Hamiltonian.
