Add a default button to the parent panel.
<asp:Panel ID="someOuterPanel" runat="server" DefaultButton="btn_Process">
<asp:TextBox id="someTextBoxOutsideInnerButtonPanel" runat="server" />
... other buttons and stuff...
<asp:Panel ID="pnl_BeneficaryDetails" runat="server" DefaultButton="btn_Process" >
<asp:ImageButton runat="server" ID="btn_Process" OnClick="btn_Process_Click" Visible="false" ImageUrl="~/image/process.png" ToolTip="Process Payment" />