The corrections to Newtonian gravity are called post-Newtonian (not to confuse with PPN or parametrized post-Newtonian expressions used to compare modified theories of gravity).
The corrections are usually done by expanding Einsteins equation into series in $\dfrac{G}{c^2}$ or $\dfrac{v}{c}$, the latter being more common, as the expnasion parameter is dimensionless ($v$ stands for the velocities of the bodies). Such an expansion effectively produces a set of corrections to the newtonian forces. The corrections proportional to $(\dfrac{v}{c})^n$ and called $nPN$ corrections. To my knowledge, one can find in the literature the expressions for the forces for up to $3.5PN$ order.
Concerning feasibility, the higher the order $n$, the more complicated are the expressions. While $1PN$ corrections can be implemented almost "for free", the following orders may be described by the equations couple of pages long. These guys talk about more or less the same problem you are doing http://labs.adsabs.harvard.edu/ui/abs/2009ApJ...695..455B, and use up to $2.5PN$ corrections.
There might be more proficient triggers to turn on/off the correction terms, but as a good one may compare directly the expansion factors $(\dfrac{v}{c})^n$ to some threshold values, where $v$ here is the relative velocity of the black holes.
There is a huge body of relevant research to your project. If you are more interested in dynamics, there are many other factors, such as collisions with stars/interstellar medium(if relevant in dwarf galaxies)/dynamic friction/etc, which probably are much more important than PN corrections during most of the galaxy merger. However, if you want to have accurate descriprion of the black hole late inspiral, say in order to extract gravitational waveforms, including PN corrections is really important.
Practically, I would trace the factor $\dfrac{v}{c}$ during everyday simulations, check its values and then decide, whether PN corrections are needed. Alternatively, search the literature on ads.
