When you open your MSI with Orca (or equivalent MSI viewer), do you see the "UAC Compliant" check box checked? Sample screenshot here:

You should really use a more flexible and capable MSI tool than the Visual Studio Installer projects. They are good for a few purposes, but lack flexibility and there are numerous other problems: summary of VS Project problems (short form).
Per-User setups considered harmful: Some words of warning against per user setups. Here is one more answer on that.
A simple per-user folder installation in WiX (insert UPPERCASE GUIDs in locations shown with "PUT-GUID-HERE" (2 occurrences) - you can use this GUID generator):
<?xml version="1.0" encoding="UTF-8"?>
<Wix xmlns="http://schemas.microsoft.com/wix/2006/wi">
  <Product Id="*" Name="PerUserSample" Language="1033" Version="1.0.0.0" Manufacturer="-" UpgradeCode="PUT-GUID-HERE">
    <Package InstallerVersion="200" Compressed="yes" InstallScope="perUser" InstallPrivileges="limited" />
    <MajorUpgrade DowngradeErrorMessage="A newer version of [ProductName] is already installed." />
    <MediaTemplate EmbedCab="yes" />
    <UIRef Id="WixUI_Mondo" />
    <Feature Id="ProductFeature" Title="PerUserSample" Level="1" />
    <Directory Id="TARGETDIR" Name="SourceDir">
      <Directory Id="AppDataFolder">
        <Directory Id="Something" Name="Something">
          <Component Feature="ProductFeature" Guid="PUT-GUID-HERE">
            <RegistryValue Root="HKCU" Key="Software\[Manufacturer]\[ProductName]\Test"
                           Name="installed" Type="integer" Value="1" KeyPath="yes"/>
            <File Source="C:\Windows\Notepad.exe" />
            <RemoveFolder Id="Something" Directory="Something" On="uninstall" />
          </Component>
        </Directory>
      </Directory>
      
      <Directory Id="ProgramFilesFolder">
        <Directory Id="INSTALLFOLDER" Name="PerUserSample" />
      </Directory>
    </Directory>
  </Product>
</Wix>