What happens when a cool brown dwarf slowly accretes matter? Let's imagine that we take a brown dwarf close to the minimum mass of a red dwarf.
Then let's allow it a few quadrillion years to cool.
Finally slowly accrete matter at a rate of say 1 Jupiter mass every 100 trillion years. How much mass would we be able to add before fusion ignites.
 A: The brown dwarf has essentially become a cold hydrogen white dwarf, supported by electron degeneracy pressure. Adding more material (at a slow rate) will make this object more massive and because it is supported by non-relativistic electron degeneracy pressure, the radius will decrease as $M^{-1/3}$ and the density will increase as $\rho \propto M^2$. At high enough densities pycnonuclear reactions will start, even in cold degenerate material. I roughly calculate this occurs when the mass is $>0.6$ solar masses.
Since degeneracy pressure is independent of temperature and you are accreting slowly and allowing plenty of time for any gravitational potential energy to be radiated away, then the temperature will never increase enough to initiate conventional thermonuclear fusion.
However, because the white dwarf is cold, the protons will arrange themselves into a crystalline lattice. At high enough densities, the zero point oscillations in the lattice to overcome the Coulomb repulsion, initiating pycnonuclear reactions (Salpeter & van Horn (1969), possibly leading to detonation of the object.
It's a bit difficult to find good numbers for the density at which pycnonuclear fusion of hydrogen in cold degenerate matter occurs, but it is something like $10^8$ to $10^9$ kg/m$^3$. Thus your question boils down to, at what mass does the central density of the brown dwarf reach these values?
Back-of-the-envelope: A low-mass carbon white dwarf has a radius of $0.01 M^{-1/3}$, where the radius and mass are in solar units. But a hydrogen white dwarf would be about 3 times bigger, because it has two times as many electrons per mass unit.
The density then of a hydrogen brown/white dwarf is roughly $5\times 10^7 M^2$ kg/m$^3$. Of course, this is the average density; the core density will be higher by a factor of 6 for a $n=1.5$ polytropic structure (Pesnell (2016).
So my rough calculation suggests if the mass is increased to somewhere between 0.6 and 2 times the mass of the Sun (note that the Chandrasekhar mass for a hydrogen white dwarf is 4 times higher than for a carbon white dwarf), then the central density will be high enough for pycnonuclear fusion to commence.
