You not at all made the case you need two buttons.

And you not at all stated why this row click data that you want to copy into the paste buffer can't be included in each row. And since it can be included in each row, then you don't need to run any server side code here.

So, let's assume a simple gridview. And say we want to click on a button on a row, and have some data from that row be copied into your paste buffer.

And VERY little reason to call server-side information UNLESS that information is not displayed in the current row and cannot be (and to be MORE specific, that data cannot be included in the row of data - even if not to be dispalyed in the gv, as long as that data is PART OF the row data source, we can get/use/enjoy that data in our client side click event).

Then REALLY to suggest that data can't be included in the current row on grid render then, right???

Let's create a simple gv. And then drop in a button and use a client side click event for this.

Our gv and markup is thus this:

And we ONLY need to drop in a plane jane regular aspnet button.

<asp:GridView ID="GridView1" 
    runat="server" CssClass="table" AutoGenerateColumns="false"
    width="42%" DataKeyNames="ID" >
    <Columns>
        <asp:BoundField DataField="FirstName" HeaderText="FirstName"  />
        <asp:BoundField DataField="LastName" HeaderText="LastName"    />
        <asp:BoundField DataField="HotelName" HeaderText="Hotel Name"    />
        <asp:BoundField DataField="Description" HeaderText="Description" ItemStyle-Width="270" />
        <asp:TemplateField HeaderText="Edit">
            <ItemTemplate>
                <asp:Button ID="cmdEdit" runat="server" Text="Copy!" 
                    CssClass="btn" 
                    MyHotelName = '<%# Eval("HotelName") %>'
                    OnClientClick="MyCopy(this);return false;"
                    />
            </ItemTemplate>
        </asp:TemplateField>
    </Columns>
</asp:GridView>

<br />

<script>
    function MyCopy(btn) {

        mytext = $(btn).attr("MyHotelName")

        navigator.clipboard.writeText(mytext).then(function () {
            console.log('Copy to clipboard was successful!');
        }, function (err) {
            console.error('Could not copy text: ', err);
        });
    }
</script>

And code behind:

    protected void Page_Load(object sender, EventArgs e)
    {
        if (!IsPostBack)
            LoadGrid();
    }

    void LoadGrid()
    {
        using (SqlConnection conn = new SqlConnection(Properties.Settings.Default.TEST4))
        {
            string strSQL = "SELECT * FROM tblHotelsA ORDER BY HotelName";

            using (SqlCommand cmdSQL = new SqlCommand(strSQL, conn))
            {
                DataTable rstData = new DataTable();
                conn.Open();
                rstData.Load(cmdSQL.ExecuteReader());
                GridView1.DataSource = rstData;
                GridView1.DataBind();
            }
        }
    }

And now we have this:

so, click on any row copy! button, and the hotel name will be in the browsers clipboard/paste buffer.

You can of course replace our custom attribute with other column values - even values that don't appear in the gv column list for display, but any valid column from the binding datasource that is being feed into the gv can be used here.

Edit: user needs PK row ID from database

Ok, so we not looking to pull/use existing data, but get the database row PK id, and then generate some key or value or walk the dog.

So, our button becomes this:

                <asp:Button ID="cmdEdit" runat="server" Text="Copy!" 
                    CssClass="btn" 
                    OnClientClick = '<%# "MyCopy(" + Eval("ID") + ");return false" %>'
                    />

And now, the js routine becomes this:

<script>
    function MyCopy(id) {
    // do a web service call, or whatever to get key value
    // code to put value into paste buffer