If I switch to use OpenJDK, will I be faced with this same limitation? That is, would I be violating a license?
There is no such limitation in the OpenJDK licensing which is a combo of CDDL and GPL with classpath exception. Run with it without fear.
As a side note, there is no good reason to use the Oracle BCL-licensed JDK. In fact reading its ever changing license terms, you can barely use it for development and that is about it.
The OpenJDK is always a better alternative IMHO.
if the class I'm modifying is covered by the "Classpath Exception" to the GPL, then does your answer change at all?
No. To the contrary: the BCL does not give any permission to modify code.
The GPL with Classpath does allow that alright. You still have to obey the GPL for the code you modify of course and -- unless you enter the grey area of derivative work -- your code using this modified code would not be subject to the GPL's requirements.