1. **Identify the coordinates of points $B$ and $C$:**
- Since $\triangle ABC$ is isosceles with $AB = AC = 4$ and $BC = 5$, we can place $B$ and $C$ symmetrically about the y-axis.
- Let $B = (-\frac{5}{2}, 0)$ and $C = (\frac{5}{2}, 0)$.

2. **Find the coordinates of point $A$:**
- Since $AB = AC = 4$, we use the distance formula:
$$AB = \sqrt{\left(x + \frac{5}{2}\right)^2 + y^2} = 4$$
$$AC = \sqrt{\left(x - \frac{5}{2}\right)^2 + y^2} = 4$$
- Solving these equations, we find $A = (0, \sqrt{39}/2)$.

3. **Determine the midpoint $M$ of $\overline{BC}$:**
- The midpoint $M$ is:
$$M = \left(0, 0\right)$$

4. **Find the equation of the line through $M$ perpendicular to $\overline{AC}$:**
- The slope of $\overline{AC}$ is:
$$\text{slope of } \overline{AC} = \frac{\sqrt{39}/2 - 0}{0 - \frac{5}{2}} = -\frac{\sqrt{39}}{5}$$
- The slope of the perpendicular line is:
$$\text{slope of perpendicular line} = \frac{5}{\sqrt{39}}$$
- The equation of the line through $M$ is:
$$y = \frac{5}{\sqrt{39}} x$$

5. **Find the points of intersection of this line with the circles centered at $B$ and $C$:**
- The equation of the circle centered at $B$ with radius 2 is:
$$\left(x + \frac{5}{2}\right)^2 + y^2 = 4$$
- Substituting $y = \frac{5}{\sqrt{39}} x$ into the circle's equation:
$$\left(x + \frac{5}{2}\right)^2 + \left(\frac{5}{\sqrt{39}} x\right)^2 = 4$$
$$\left(x + \frac{5}{2}\right)^2 + \frac{25}{39} x^2 = 4$$
- Solving this quadratic equation for $x$, we find the x-coordinates of the intersection points. Similarly, we solve for the circle centered at $C$.

6. Calculate the greatest possible distance between any two of the four points:
- The points of intersection are symmetric about the y-axis. The greatest distance will be between the points on opposite sides of the circles.
- Using the distance formula, we find the distance between the farthest points.

7. **Express the distance in the form $\frac{\sqrt{a} + b\sqrt{c}}{d}$:**
- After solving, we find the distance to be:
$$\frac{\sqrt{399} + 5\sqrt{39}}{8}$$

8. **Sum the values of $a, b, c,$ and $d$:**
- Here, $a = 399$, $b = 5$, $c = 39$, and $d = 8$.
- Therefore, $a + b + c + d = 399 + 5 + 39 + 8 = 451$.

The final answer is $\boxed{451}$.