How to bind data to a custom Windows Form using SplitContainer?

Question

I've created a form that I call a ListChooser. It looks like so:
It consists of a System.Windows.Forms.Button and .TextBox.

Clicking on the Button (shown above with the text "Fruits") causes a dialog to popup and some selections are made and the TextBox is populated, like so:
My problem is I can't figure out how to set a System.Windows.Forms.Binding to this TextBox. I think the complication is that it's not just a TextBox, it's 2 components in one. (I know how to data bind with just a standalone TextBox). If I go to the TextBox > Properties > DataBindings > Advanced > ..., All entries listed under Property are not selectable:

I've tried this which has no effect:

private System.Windows.Forms.BindingSource myBindingSource;
this.myBindingSource = new System.Windows.Forms.BindingSource(this.components);
this.myBindingSource.DataSource = typeof(Foods) // Fruits is a Property of Foods;
myListChooser.DataBindings.Add(new System.Windows.Forms.Binding("Text", this.myBindingSource, "Fruits", true));

How do I do this?

ListChooser.Designer.cs:

partial class ListChooser: UserControl
{
    /// <summary> 
    /// Required designer variable.
    /// </summary>
    private System.ComponentModel.IContainer components = null;

    /// <summary> 
    /// Clean up any resources being used.
    /// </summary>
    /// <param name="disposing">true if managed resources should be disposed; otherwise, false.</param>
    protected override void Dispose(bool disposing)
    {
        if (disposing && (components != null))
        {
            components.Dispose();
        }
        base.Dispose(disposing);
    }

    #region Component Designer generated code

    /// <summary> 
    /// Required method for Designer support - do not modify 
    /// the contents of this method with the code editor.
    /// </summary>
    private void InitializeComponent()
    {
        this.btnPopup = new System.Windows.Forms.Button();
        this.textBox = new System.Windows.Forms.TextBox();
        this.splitContainer1 = new System.Windows.Forms.SplitContainer();
        this.splitContainer1.Panel1.SuspendLayout();
        this.splitContainer1.Panel2.SuspendLayout();
        this.splitContainer1.SuspendLayout();
        this.SuspendLayout();
        // 
        // btnPopup
        // 
        this.btnPopup.AutoSize = true;
        this.btnPopup.Dock = System.Windows.Forms.DockStyle.Fill;
        this.btnPopup.Location = new System.Drawing.Point(0, 0);
        this.btnPopup.Margin = new System.Windows.Forms.Padding(0);
        this.btnPopup.Name = "btnPopup";
        this.btnPopup.Size = new System.Drawing.Size(75, 21);
        this.btnPopup.TabIndex = 0;
        this.btnPopup.Text = "Type";
        this.btnPopup.TextAlign = System.Drawing.ContentAlignment.MiddleLeft;
        this.btnPopup.UseVisualStyleBackColor = true;
        this.btnPopup.Click += new System.EventHandler(this.btnPopup_Click);
        // 
        // txtList
        // 
        this.textBox.Anchor = ((System.Windows.Forms.AnchorStyles)(((System.Windows.Forms.AnchorStyles.Top | System.Windows.Forms.AnchorStyles.Left)
                    | System.Windows.Forms.AnchorStyles.Right)));
        this.textBox.BackColor = System.Drawing.SystemColors.Window;
        this.textBox.Font = new System.Drawing.Font("Microsoft Sans Serif", 8.25F, System.Drawing.FontStyle.Regular, System.Drawing.GraphicsUnit.Point, ((byte)(0)));
        this.textBox.Location = new System.Drawing.Point(0, 0);
        this.textBox.Name = "txtList";
        this.textBox.ReadOnly = true;
        this.textBox.Size = new System.Drawing.Size(297, 20);
        this.textBox.TabIndex = 1;
        this.textBox.TabStop = false;
        this.textBox.TextChanged += new System.EventHandler(this.txtList_TextChanged);
        //this.textBox.Leave += new System.EventHandler(this.txtList_Leave);
        // 
        // splitContainer1
        // 
        this.splitContainer1.Dock = System.Windows.Forms.DockStyle.Fill;
        this.splitContainer1.FixedPanel = System.Windows.Forms.FixedPanel.Panel1;
        this.splitContainer1.Location = new System.Drawing.Point(0, 0);
        this.splitContainer1.Margin = new System.Windows.Forms.Padding(0);
        this.splitContainer1.Name = "splitContainer1";
        // 
        // splitContainer1.Panel1
        // 
        this.splitContainer1.Panel1.Controls.Add(this.btnPopup);
        // 
        // splitContainer1.Panel2
        // 
        this.splitContainer1.Panel2.Controls.Add(this.textBox);
        this.splitContainer1.Size = new System.Drawing.Size(375, 21);
        this.splitContainer1.SplitterDistance = 75;
        this.splitContainer1.SplitterWidth = 3;
        this.splitContainer1.TabIndex = 2;
        // 
        // ListChooser
        // 
        this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 13F);
        this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
        this.BackColor = System.Drawing.Color.WhiteSmoke;
        this.Controls.Add(this.splitContainer1);
        this.MaximumSize = new System.Drawing.Size(1200, 21);
        this.Name = "ListChooser";
        this.Size = new System.Drawing.Size(375, 21);
        this.splitContainer1.Panel1.ResumeLayout(false);
        this.splitContainer1.Panel1.PerformLayout();
        this.splitContainer1.Panel2.ResumeLayout(false);
        this.splitContainer1.Panel2.PerformLayout();
        this.splitContainer1.ResumeLayout(false);
        this.ResumeLayout(false);

    }

    #endregion

    private System.Windows.Forms.Button btnPopup;
    private System.Windows.Forms.TextBoxBase textBox;

    private SplitContainer splitContainer1;
}