Out of the box, the default Rich Text Editor profile can be set via the “HtmlEditor.DefaultProfile” setting.
<setting name="HtmlEditor.DefaultProfile" value="/sitecore/system/Settings/Html Editor Profiles/Rich Text Default"/>
Unfortunately, this only allows you to set the default profile for the entire Sitecore instance making it the default profile for each site.
Set it up for Multisite
Luckily, it is possible to create a custom EditorConfiguration which reads an attribute from Site Settings and replaces the default profile:
Add the following config file and set correct namespaces for HtmlEditor.DefaultConfigurationType:
Edit sites config and point to your preferred rich text editor profile:
<site name="mySite" defaultRichTextProfile="/sitecore/system/Settings/Html Editor Profiles/Rich Text Full" ...
Tested on 8.0 and 8.1.
UPDATE 04.03.2016: Changed code to only override the default profile. If a different profile was defined at field level it will not be overridden.