Is there a way to load the content types using DrupalWebTestCase?
Try the SimpleTest Clone module.

SimpleTest performs the test on a clean Drupal installation, so if you have created Content Types they will not be available.

SimpleTest Clone craetes a copy of the database (using a table prefix) and performs the tests over those tables, so all config you have done is present.

Once the tests are donde the extra tables are deleted.

The drawback fo this method is that is a little slower than standard SimpleTest because the table copy operation.

To use SimpleTest Clone just subclass your test class form SimpleTestCloneTestCase class instead of DrupalWebTestCase class.