AWTUtilities, as the package implies is an implementation package and is subject to change. I don't think it is a good idea to distribute the class from both technical and legal viewpoints.
Technically, com.sun.awt.AWTUtlities is bound to have possibly unknown dependencies on other classes and internal implementation details, specific to Java 6 u10 and above (the release in which it appeared). This unknown dependency angle is bound to hold water, since painting and graphics will require changes in some of the native implementations as well.
Secondly, this is bound to change in Java 7, since the only release Sun hasn't made a release of java.awt.AWTUtilities with support for transparency, is because they do not make changes to public APIs except in major releases.
IANAL, but I do not think it is wise to engage in the act of redistributing software without having run past a lawyer. Besides, customers do not like the idea of having an unsupported deployment of any software in their systems.
Update
All Sun JREs (not just JDKs) from 6u10 onwards come with com.sun.awt.AWTUtilities, so the simplest course of action would be to get your users to have Java 6u10 or later on their systems, and for your application to handle any resulting exception gracefully.