I have the basic code for the inverse kinematics of the robot working:

import math
from machine import I2C, Pin
import servo


PI2 = math.pi / 2
PI3 = math.pi / 3


class Robot:
    ARM_LENGTH = 42
    ROD_LENGTH = 134
    HUB_LENGTH = 27
    BASE_LENGTH = 62

    BASE_X = (
        0,
        (BASE_LENGTH - HUB_LENGTH) * math.sin(2 * PI3),
        (BASE_LENGTH - HUB_LENGTH) * math.sin(4 * PI3),
    )
    BASE_Y = (
        BASE_LENGTH - HUB_LENGTH,
        (BASE_LENGTH - HUB_LENGTH) * math.cos(2 * PI3),
        (BASE_LENGTH - HUB_LENGTH) * math.cos(4 * PI3),
    )

    def __init__(self):
        self.i2c = I2C(-1, Pin(5), Pin(4))
        self.servos = servo.Servos(self.i2c, min_us=700, max_us=2400,
                                   degrees=180)
        self.trims = (0, -0.1, -0.2)

    def home(self):
        self.move(0, 0)
        self.move(1, 0)
        self.move(2, 0)

    def move(self, arm, radians):
        radians += self.trims[arm] + PI2
        self.servos.position(arm, radians=radians)

    def arm_ik(self, distance):
        alpha = math.acos(
            (self.BASE_LENGTH - self.HUB_LENGTH) /
            (distance)
        )
        beta = math.acos(
            (distance ** 2 + self.ARM_LENGTH ** 2 - self.ROD_LENGTH ** 2) /
            (2 * self.ARM_LENGTH * distance)
        )
        return math.pi - alpha - beta

    def robot_ik(self, x, y, z):
        return tuple(math.sqrt(
            (self.BASE_X[arm] - x) ** 2 +
            (self.BASE_Y[arm] - y) ** 2 +
            z ** 2
        ) for arm in (0, 1, 2))

    def move_to(self, x, y, z):
        for arm, distance in enumerate(self.robot_ik(x, y, z)):
            self.move(arm, self.arm_ik(distance))

As you can see, shifting the servo by the width of the hub greatly simplified the equations. I am pretty happy with how it works -- managed to draw a circle on a piece of paper, and the circle is actually round. One thing is strange, though -- the circle was supposed to have 5cm radius, but it's around 2cm instead... I might have an error there somewhere. I will keep experimenting with it.