The Dr. Seuss name alone is not what's trademarked. The specific appearance of the name, with all the characteristics of the font and style is trademarked, as is the appearance of the Cat in the Hat.
A name alone is not enough to get a trademark. And logically you can see the problem of copyrightiing a name. If "Benjamin Franklin" was copyrighted, then it would be illegal to name your kid Benjamin Franklin, even if your last name was Franklin and your dad's name was Benjamin.
What I would do is copyright a work under the pseudonym, which at least gives you some legal standing. Also, if you sign any contracts as an author, have wording included about the use of the pseudonym (e.g., can be used to publicize the book, but cannot be used as an author's name for books you did not write). A lot of authors license the use of their name (Tom Clancy, for one) for series of books. Also, a lot of publishers use a fictional pseudonoymous author to publish a series by several ghostwriters.
To get the trademark, you could write the name in a particular style (big fancy script, designed just for you) and trademark that. At least then you have legal standing to argue against others using it. But that would only work if your works appear under that stylized name.
