Custom Editor Configuration allowing you to set RichText Editor Profiles per Site instead of using the global "HtmlEditor.DefaultProfile" setting.

MultiSiteRichTextEditorProfile.config

<?xml version="1.0" encoding="utf-8" ?>
<configuration xmlns:patch="http://www.sitecore.net/xmlconfig/">
  <sitecore>
    <settings>

      <!-- Set custom RichText Configuration Type to allow multi site configuration -->
      <setting name="HtmlEditor.DefaultConfigurationType" value="MyProject.MultiSiteRichTextConfiguration,MyProject"/>
      <setting name="MultiSiteRichTextConfiguration.DefaultDatabase" value="master"/>
      <setting name="MultiSiteRichTextConfiguration.CoreDatabase" value="core"/>

    </settings>
  </sitecore>
</configuration>

MultiSiteRichTextEditorProfile.cs

/// <summary>
/// Allows setting a default rich text profile per Site. 
/// </summary>
public class MultiSiteRichTextConfiguration : EditorConfiguration
{
    public MultiSiteRichTextConfiguration(Item profile) : base(GetSiteProfile(profile))
    {
    }

    /// <summary>
    /// Returns the rich text editor profile allowing it to be overridden by site settings
    /// </summary>
    /// <param name="profile"></param>
    /// <returns></returns>
    private static Item GetSiteProfile(Item profile)
    {
        // Don't override if a profile other than default is used
        if (!IsDefaultProfile(profile))
        {
            return profile;
        }
        
        var item = GetItemFromUri();

        var site = GetSiteFromItem(item);

        // Override profile 
        var siteProfile = site.Properties["defaultRichTextProfile"];
        if (!string.IsNullOrEmpty(siteProfile))
        {
            return Factory.GetDatabase(Settings.GetSetting("MultiSiteRichTextConfiguration.CoreDatabase", "core")).GetItem(siteProfile);
        }

        return profile;
    }
    
    /// <summary>
    /// Returns true if the profile is the same as the default profile specified in HtmlEditor.DefaultProfile
    /// </summary>
    /// <param name="profile"></param>
    /// <returns></returns>
    private static bool IsDefaultProfile(Item profile)
    {
        return Settings.GetSetting("HtmlEditor.DefaultProfile", string.Empty).Equals(profile.Paths.FullPath, StringComparison.CurrentCultureIgnoreCase);
    }

    /// <summary>
    /// Returns the most applicable site for an item
    /// </summary>
    /// <param name="item"></param>
    /// <returns></returns>
    public static SiteInfo GetSiteFromItem(Item item)
    {
        var siteInfoList = Factory.GetSiteInfoList();

        SiteInfo currentSiteinfo = null;
        var matchLength = 0;
        foreach (var siteInfo in siteInfoList)
        {
            if (item.Paths.FullPath.StartsWith(siteInfo.RootPath, StringComparison.OrdinalIgnoreCase) && siteInfo.RootPath.Length > matchLength)
            {
                matchLength = siteInfo.RootPath.Length;
                currentSiteinfo = siteInfo;
            }
        }

        return currentSiteinfo;
    }

    /// <summary>
    /// Returns the context item from the current query string
    /// </summary>
    /// <returns></returns>
    private static Item GetItemFromUri()
    {
        ItemUri itemUri = ItemUri.ParseQueryString();

        var database = Factory.GetDatabase(Settings.GetSetting("MultiSiteRichTextConfiguration.DefaultDatabase", string.Empty));

        if (database == null)
            return null;

        return database.GetItem(itemUri.ItemID);
    }
}

Use the following property on your site setting and point to the RTE Profile you want to set as default:
defaultRichTextProfile="/sitecore/system/Settings/Html Editor Profiles/Rich Text Full"