Since you can see the reportviewer for forms: please note that you select Microsoft.Reporting.WebForms for asp.net.
Do you see the tab 'Reporting' in your toolbox? This should contain MicrosoftReportViewer. If not, try reinstalling the redistributable(s).
But do you really need it in your toolbox? You can also add the following line to your page (you can change the version also to 10.0.0.0).
<%@ Register assembly="Microsoft.ReportViewer.WebForms, Version=18.104.22.168, Culture=neutral, PublicKeyToken=b03f5f7f11d50a3a" namespace="Microsoft.Reporting.WebForms" tagprefix="rsweb" %>
And the actual reportviewer:
<rsweb:ReportViewer ID="ReportViewer1" runat="server" Font-Names="Verdana" Font-Size="8pt" ShowRefreshButton="False" Width="790px">
If this works you at least have a work-around.