The scenario you described is different from what I initially assumed. The "Basic Page" is actually a web part page with only one web part - the default content editor web part. Naturally, it is not cross-browser compatible and you want to replace it. This is possible through the SharePoint Designer - Open the site with Designer, expand the Document Library (e.g. Shared Documents) and open the page you created. You can then add a new radEditor web part by clicking the Title Bar web part zone and selecting it from the Server Gallery. After you add the editor web part you can replace the default content editor web part with ours.
We tried this scenario but there was one drawback - the "edit content" link on the page is hardwired to the default content editor, so after we replaced it with our web part, the link no longer worked. You could still edit the web part and save its content through the right web part menu.
Overall, our conclusion is that it is possible to replace the editor in the "Basic Page" scenario, however, the user experience will be different so you need to decide whether the solution we offered is good for you or not.