请尝试使用：

MDApp.get_running_app().root.get_screen('screenTwo').ids.my_label.text = myResult

这会取得MDApp并指涉它的root（WindowManager），以取得所需的Screen。

我不确定拥有独立类的实用性--但是试着在这里给出一些传递信息的例子。我认为关键是在你的主应用程序中设置对象引用。
main.py

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from kivy.uix.screenmanager import ScreenManager, Screen
from kivymd.app import MDApp
from kivymd.uix.label import MDLabel
from kivymd.uix.textfield import MDTextField
from kivy.lang import Builder

class Screen1Main(Screen):
    kv_text_field: MDTextField

class Screen2(Screen):
    my_label: MDLabel

    def __init__(self, **kwargs):
        super().__init__(**kwargs)

class myExternalClass:

    def foo2(self, arg):
        self.myResult = "anything"
        self._screen2.my_label.text = arg

    def __init__(self, screen2):
        self.myResult = ""
        # get a reference to this screen so you can access it in this class instance
        self._screen2: Screen2 = screen2

class WindowManager(ScreenManager):
    pass

class MainApp(MDApp):

    def foo1(self, my_button):
        # 1. do something
        myArg = "from main App"
        print(f"{my_button}, {type(my_button)}")

        # 2. then call foo2()
        self.my_external_class.foo2(myArg)

    def main_screen(self, my_button):
        print(my_button)
        self.my_top_widget.current = "Screen1Main"

    def kv_change_screen(self, my_button, arg: str):
        print(f"{self}, {my_button}, {arg}")
        self.screen2.my_label.text = arg
        self.my_top_widget.current = "Screen2"

    def build(self) -> WindowManager:

        self.my_top_widget.add_widget(self.screen1main)
        self.my_top_widget.add_widget(self.screen2)
        self.my_top_widget.current="Screen1Main"
        return self.my_top_widget

    def __init__(self, **kwargs):
        self.title = "My Application"
        super().__init__(**kwargs)
        self.my_top_widget = WindowManager()
        self.screen2 = Screen2(name="Screen2", )
        self.screen1main = Screen1Main(name="Screen1Main")
        self.screen1main.kv_text_field.text = "message to screen 2"
        self.my_external_class = myExternalClass(screen2=self.screen2)

if __name__ == "__main__":
    # load the file by name - this is more clear
    Builder.load_file("main_app.kv")
    my_app = MainApp()
    my_app.run()

主应用程序.kv

#:import kivy kivy
# main_app.kv
<Screen1Main>:
    kv_text_field: kv_text_field
    MDBoxLayout:
        orientation: 'vertical'
        MDTextField:
            id: kv_text_field
        MDIconButton:
            text: "button"
            icon: "message-arrow-right"
            on_press: app.kv_change_screen(self, kv_text_field.text)

<Screen2>:
    my_label: my_label
    GridLayout:
        cols: 2
        Label:
            text: "Screen2"
        MDLabel:
            id: my_label
            text: "-"

        MDFlatButton:
            id: my_button
            text: "foo1"
            on_release: app.foo1(self)
        MDIconButton:
            id: my_button
            icon: "message-arrow-left"
            on_release: app.main_screen(self)

您可能不需要WindowManager的类定义，如果不打算扩充该类，则可以直接使用ScreenManager。
我希望这里的例子可以帮助你，即使我没有直接遵循你的应用程序的确切流程和结构。