You could edit the style you are using to achieve this.
In EndNote you can go to Edit > Output Styles > Open Style Manager > locate the style you are using > Click Edit > go to Templates under Citations > here you would see a second format under Citation - Author (Year) > so you can delete the current information then go to the Insert Field button in the upper right and select Bibliography Number > you can then add the ( ) around Bibliography Number so that you have (Bibliography Number) > go To File > Save As > (EndNote will not allow you to change the existing style in EndNote but will always apply the change to a copy of the style) Click Save to save the changes to a copy of the style > close the style window.
You can the insert the citation as you would normally then use the Edit and Manage Citation command to change the Formatting option at the bottom from Default to Author (Year) then click OK.
You can try the attached file to test this.
Save the attachment from this post to your desktop
Double-click on the downloaded file to open it in the EndNote software.
Click the File menu in EndNote and select Save As.
Click Save.
Close the style window.
You can now delete the file on the desktop the style has been added to EndNote.