To find the minimum value of |2z - 6 + 5i|, we first need to understand the geometric interpretation of the given inequality |z - 3 - 2i| ≤ 2.

Geometric Interpretation:
- The inequality |z - 3 - 2i| ≤ 2 represents the set of all complex numbers that lie within or on the boundary of a circle centered at the point (3, 2) in the complex plane, with a radius of 2 units.
- This circle can be represented as C((3, 2), 2), where C denotes a circle, (3, 2) represents the center, and 2 is the radius.

Finding the Minimum Value:
- To find the minimum value of |2z - 6 + 5i|, we can consider the transformation of the given circle under the transformation w = 2z - 6 + 5i.
- Let's denote the transformed circle as C', which is defined by the equation |w| ≤ r', where r' is the radius of the transformed circle.
- We need to find the minimum value of r' such that the transformed circle C' intersects or just touches the origin (0, 0) in the complex plane.
- If the transformed circle C' touches the origin, then the minimum value of |2z - 6 + 5i| will be equal to r'.
- Let's solve for the value of r':

Transformation of the Circle:
- Substituting w = 2z - 6 + 5i, we have |2z - 6 + 5i| ≤ r'.
- Simplifying, we get |2z - (6 - 5i)| ≤ r'.
- Comparing this with the original circle equation, we find that the transformed circle C' has a center at (3 - 5i) and a radius of r'.

Finding the Minimum Value of the Radius:
- To find the minimum value of r', we need to find the distance between the center of C' and the origin (0, 0) in the complex plane.
- The distance between two complex numbers a + bi and c + di is given by |(a + bi) - (c + di)| = |(a - c) + (b - d)i|.
- Applying this formula, the distance between the center of C' (3 - 5i) and the origin (0, 0) is |3 - 5i|.
- Therefore, the minimum value of |2z - 6 + 5i| is equal to |3 - 5i|.

Calculating the Minimum Value:
- The distance between two complex numbers a + bi and c + di is given by |(a + bi) - (c + di)| = √((a - c)^2 + (b - d)^2).
- Applying this formula, we have |3 - 5i| = √((3 - 0)^2 + (-5 - 0)^2) = √(9 + 25) = √34.
- Hence, the minimum value of |2z - 6 + 5i| is √34, which is approximately equal to 5.

Therefore, the correct answer is '5'.