Taking the Pascal approach above, I went looking for the right way to create a SchemaUpdate object in Spring and realized that I didn't need it. Instead, I can just get the Spring sessionFactory object for Hibernate and ask it to delete / create the schema. Combining this with the rest of Pascal's solution, we get the following:
LocalSessionFactoryBean localSessionFactoryBean = (LocalSessionFactoryBean)appContext.getBean("&sessionFactory");
localSessionFactoryBean.dropDatabaseSchema();
localSessionFactoryBean.createDatabaseSchema();
Cache cache = sf.getCache();
cache.evictEntityRegions();
cache.evictCollectionRegions();
cache.evictQueryRegions();
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