Since $α$ is an obtuse angle and $\sin(α + \frac{π}{3}) = -\frac{4}{5}$,

We know that $π < α + \frac{π}{3} < \frac{3π}{2}$,

This implies $\cos(α + \frac{π}{3}) = -\frac{3}{5}$,

Now, let's find $\sin 2(α + \frac{π}{3}) = 2\sin(α + \frac{π}{3})\cos(α + \frac{π}{3}) = 2 \times (-\frac{4}{5}) \times (-\frac{3}{5}) = \frac{24}{25}$,

And, $\cos 2(α + \frac{π}{3}) = 2\cos^2(α + \frac{π}{3}) - 1 = 2 \times (-\frac{3}{5})^2 - 1 = -\frac{7}{25}$,

Now, we need to find $\cos(2α + \frac{5π}{12}) = \cos[(2α + \frac{2π}{3}) - \frac{π}{4}]$,

Using the cosine difference formula, we get $\cos(2α + \frac{5π}{12}) = \cos(2α + \frac{2π}{3})\cos\frac{π}{4} + \sin(2α + \frac{2π}{3})\sin\frac{π}{4}$,

Substituting the values, we get $\cos(2α + \frac{5π}{12}) = -\frac{7}{25} \times \frac{\sqrt{2}}{2} + \frac{24}{25} \times \frac{\sqrt{2}}{2} = \boxed{\frac{17\sqrt{2}}{50}}$.

This problem involves the application of trigonometric functions of the same angle and trigonometric identities transformation. It requires flexible use of formulas.