I decided to start working on my own answer after I asked the question, so here's the result of a few days' work.
An excellent (and very recent) paper regarding accretion is Debnath (2015), which can be applicable at least for material gathered onto the surface of the red dwarf.
Debnath assumes a static1, spherically symmetric metric:
$$ds^2=-A(r)dt^2+\frac{1}{B(r)}dr^2+r^2(d \theta^2+ \sin \theta d \phi^2) \tag{1}$$
which uses the (-,+,+,+) sign convention. For now, we can leave $A$ and $B$ undetermined functions of $r$. We have to treat the surrounding matter as a perfect fluid, with a stress-energy tensor of
$$T_{\mu \nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu \nu} \tag{2}$$
with $\rho$ and $p$ being the density and pressure, respectively.2 $u_{\alpha}$ is the four-vector, with the condition that $u_{\alpha}u^{\alpha}=-1$. For this fluid, though,
$$u^{\alpha}=(u^0,u^1,0,0)$$
We can then re-write the earlier condition as
$$g_{00}u^0u^0+g_{11}u^1u^1=-1$$
Substituting in that $g_{00}=g_{tt}=A(t)$ and $g_{11}=g_{rr}=\frac{1}{B(t)}$, as well as assuming (for simplicity) that $u^1=u$, we get
$$\left(u^0\right)^2=\frac{\left(u^1\right)^2+B}{AB} \to u_0=g_{00}u^0=\sqrt{\frac{A(u^2+B)}{B}}$$
We can also calculate $\sqrt{-g}=\sqrt{\frac{A}{B}}r^2 \sin \theta$.
The law of conservation of energy states that $u{_\mu}T_{; \nu}^{\mu \nu}=0$.3 Putting this together with $(2)$ gives
$$u^{\mu} \rho_{, \mu}+(\rho+p)u_{; \mu}^{\mu}=0$$
Doing it out, we find that
$$C=-ur^2M^{-2}\sqrt{\frac{A}{B}} \exp \left[\int_{\rho_{\infty}}^{\rho_R} \frac{1}{\rho + p(\rho)} d \rho\right] \tag{3}$$
where $\rho_R$ is the density at the radius of the red dwarf and $C$ is a constant (which will be used later).
The rate of change of mass of the black hole, $\dot{M}$ (the negative rate of change of the mass of the fluid) is expressed as4
$$\dot{M}=\int T_0^1dS \tag{4}$$
where
$$dS=\sqrt{-g}d \theta d \phi$$
From $(2)$, we get
$$\dot{M}=4 \pi CM^2(\rho + p) \tag{5}$$
Abramowicz & Fragile (2013) give a slightly different expression in place of $(4)$ (Equation 125):
$$\dot{M} = \int \sqrt{-g} \rho u^r d \theta d \phi \tag{6}$$
and use $(5)$ for the energy flux. Both expressions are applied to jets on black hole accretion disks.
Working off of Debnath, the total mass transferred to the red dwarf's surface is
$$M=\int_{t_0}^{t_f} \left[\int \sqrt{-g} \rho u^r d \theta d \phi \right] dt \tag{7}$$
where $t_0$ and $t_f$ are the initial and final times during which the red dwarf accretes material.
I haven't quite figured out the full Roche lobe calculations just yet, but I was able to find some of the major equations. Paczynski (1971) mentions that the radius of the Roche lobe of the red dwarf is
$$r_1=\left[\frac{2}{3^{4/3}} \left(\frac{M_1}{M_1+M_2} \right)^{1/3} \right]A \tag{8}$$
where $M_1$ is the mass of the red dwarf and $M_2$ is the mass of the B-type star, and $A$ is the distance between them.
The issue is that this is typically applied in binary systems, while, presumably, the red dwarf is traveling at a speed greater than the B-type star's escape velocity. It is, therefore, not orbiting it. So I'm not sure if the formula is valid.
Let's say that the civilization places a planet-sized object into the disk, inserting it at an orbital velocity $V_0$. It would then undergo Bondi accretion, as shown in Bondi (1951).5 In that paper, he goes from the expressions derived by Hoyle & Littleton and Bondi & Hoyle to get the accretion rate of
$$\dot{M} = 2 \pi (GM)^2 (v^2 + c_s^2)^{-3/2} \rho \tag{9}$$
where $v$ is relative to the fluid. Taking the limit as $v \to 0$ gives the approximation shown on Wikipedia, although it differs by a factor of 2.
We can't just use this, though, because there are other things to consider. First, all the gas and dust in the disk is orbiting at the same rate as this object, so $V_0 \neq v$. Second, the conditions change. For each orbit the object makes, the density of the matter in the path through the disk changes, because it has been swept up. Finally, the object may be severely affected by Stokes drag.
The density issue can be dealt with by simply assigning the object a number of orbits $n$ at time $t$, and saying that during each orbit, it accretes $x$ percent of the gas and dust in its way. Once this is known, an expression can be derived for the accretion during each orbit.
The Stokes drag is slightly more interesting. As shown in a derivation by Gavnholt et. al. (2004), the formula is
$$D=6 \pi \mu U a \left(1+\frac{3Re}{8} \right) \tag{10}$$
where $U=v$, $a$ is the object's radius, $\mu$ is the viscosity and $Re$ is the Reynolds number. This means that
$$\frac{dv}{dt} \propto v$$
Knowing that, and placing the object in a circular orbit such that
$$F_g=F_c \to G\frac{M_sm_o}{r^2}=\frac{mv^2}{r}$$
where $M_s$ is the mass of the B-type star, $m_o$ is the mass of the object and $r$ is the distance between them. we can write $v$ as a function of time and then solve for $r$ as a function of $v$, eventually witnessing orbital decay. Also, if $\rho$ is a function of $r$, we can further complicate everything. This also goes for the accretion experienced by the red dwarf.
I feel bad about not doing any actual calculations (i.e. with actual numbers), so I'll discuss a special case here: A dust disk surrounding a spherically symmetric body.
In a dust solution, $p=0$, so our generic equation of state $p=p(\rho)$ goes to 0. Imposing spherically symmetry means that $A=B$. Accounting for all this turns $(3)$ into
$$C=-r^2uM^{-2}\sqrt{\frac{1}{1}} \exp \left[ \int_{\rho_{\infty}}^{\rho_R} \frac{1}{\rho + 0} d \rho \right]$$
$$=-r^2uM^{-2} \exp \left[ \ln \frac{\rho_R}{\rho_{\infty}} \right]$$
$$=-r^2uM^{-2} \frac{\rho_R}{\rho_{\infty}} $$
Plugging this into $(5)$ gives us
$$\dot{M}=-4 \pi r^2u \frac{\rho_R}{\rho_{\infty}} \rho$$
Just take a given $\rho$, pick a velocity, and solve for $\dot{M}$.
I still haven't put in any numbers, but it's at the point where you don't have to do much to find the result.
Accretion by planetary-mass objects in debris disk has been observed, such as in Epsilon Eridani's debris disk (Greaves et. al. (2005); explored also by Backman et. al. (2008)). A good overview of the process is given by Janson et. al. (2013), while it was simulated by Stark & Kuchner (2009) and Nesvold & Kuchner (2014). The only issue now is to establish whether or not a Type II civilization could build such an object.
Footnotes
1This means we have to neglect rotation, which could be a problem.
2Were we to assume vanishing pressure, as in a true dust solution, things could get simpler (and, perhaps, more interesting). For now, though, we'll treat it as a perfect fluid, and treat it as homogenous.
3I'm using the convention in which a comma indicates a partial derivative and a semicolon indicates a covariant derivative.
4Raising and lowering indices via the metric tensor.
5In "thin disk" scenarios, the red dwarf might not undergo spherical accretion.
References
Abramowicz, M. A. and Fragile, P. C. "Foundations of Black Hole Accretion Disk Theory" (2013)
Backman, D. et. al. "Epsilon Eridani’s Planetary Debris Disk:
Structure and Dynamics based on
Spitzer and CSO Observations" (2008)
Bondi, H. "On Spherically Symmetric Accretion" (1951)
Debnath, U. "Accretion and Evaporation of Modified Hayward Black Hole" (2015)
Gavnholt, J. et. al. "Calculations of the Flow
Around a Sphere in a Fluid" (2004)
Janson, M. et. al. "The SEEDS Direct Imaging Survey for Planets and Scattered Dust Emission
in Debris Disk Systems" (2013)
Nesvold, E. R. and Kuchner, M. J. "Gap Clearing by Planets in a Collisional Debris Disk" (2014)
Paczynski, B. "Evolutionary Processes in Close Binary Systems" (1971)
Stark, C. C. and Kuchner, M. J. "A New Algorithm for Self-Consistent 3-D Modeling of Collisions
in Dusty Debris Disks" (2009)
