ExcludeMasses is an option of Apart2. It
allows to specify masses w.r.t which partial fraction decomposition
should not be performed,
e.g. ExcludeMasses->{m1,m2,3}.
Apart2[FAD[k, {k, m1}, {k, m2}]] // Expand\frac{1}{\text{m1}^2 \left(\text{m1}^2-\text{m2}^2\right) \left(k^2-\text{m1}^2\right)}-\frac{1}{\text{m1}^2 \left(\text{m1}^2-\text{m2}^2\right) \left(k^2-\text{m2}^2\right)}-\frac{1}{\text{m1}^2 \;\text{m2}^2 \left(k^2-\text{m2}^2\right)}+\frac{1}{k^2 \;\text{m1}^2 \;\text{m2}^2}
Apart2[FAD[k, {k, m1}, {k, m2}], ExcludeMasses -> m2] // Expand\frac{1}{\text{m1}^2 \left(k^2-\text{m1}^2\right).\left(k^2-\text{m2}^2\right)}-\frac{1}{\text{m1}^2 k^2.\left(k^2-\text{m2}^2\right)}