Per ESRI technical support, no, you cannot switch licenses mid-stream. This holds true for version 9.3.1 as well as 10 (and I'm assuming prior versions as well).
On a related note, this holds true for ArcMap and ArcCatalog, as well. You'd need to exit those applications if you want to switch from an ArcView to an ArcInfo license.
UPDATE: ESRI called me back a few minutes after hanging up with me to tell me that the first tech was wrong and that you can switch licenses mid-stream in Engine (though not when using ArcMap). However, the information the second tech provided (actually, she was an SDK programmer) is exactly the same steps I was using to try to switch licenses, so, again, this appears to be something that is either broken or doesn't work as they had intended. I'll submit a bug report and see if anything comes of it.
Final update: Per the documentation, it is not possible to reinitialize an application with a new license:
Initialize the application with the product license
Once it has been
established that the appropriate
product and extension licenses are
available, initialize the application
with the product license. Once
initialized, it is not possible to
reinitialize the application.
This is not a bug, this is expected behavior. And unfortunate.