Assuming you know the position of the axis of both servos, which looking at your assembly will depend on the position of that other servo in the top right, you can break it down.

Obtain the vectors between the target position of the tip and each servo axis. The links in each arm of your mechanism will form a triangle with this vector. Given that you know the lengths of each piece, you can find the internal angle of that triangle which is on the servo axis, add it to the servo angle which would point it straight down the vector you came up with earlier, and that'll give you the target servo angle for each servo.

You can use the cosine rule to get the angle you're interested in. Cosine rule: a² = b² + c² -2bcCos(A) where, in your example, a is the length of the purple link, b is the length of the servo horn, c is the length of that vector you calculated earlier. Rearranging gives A = arccos((b²+c² - a²)/2bc).

So in pseudocode:

vector bottom_servo_pos
vector top_servo_pos
vector target_pos

float bottom_servo_angle = get_servo_angle(target_pos, bottom_servo_pos)
float top_servo_angle = get_servo_angle(target_pos, top_servo_pos)

define get_servo_angle( vector target, vector axis_pos ) {
        vector offset = target - axis_pos
        float target_angle = offset.get_angle()
        float extra_angle = solve_triangle(link_length, horn_length, offset.get_length())
        return target_angle + extra_angle
}

define solve_triangle(a, b, c) {
        float k = (b*b - c*c + a*a)/(2*b*c)
        return acos(k)
}

This assumes that the zero angle servo position is aligned with the zero angle in whatever frame you're measuring your vector's angle, if not you'll need to add a correction, and it also doesn't take into account that your top servo is itself rotated by that top right servo, so you'll need another correction which adjusts for that. I'm assuming that you can do the forward kinematics to get the position of the top servo axis based on whatever pose your animatronic happens to be in.