I'm not sure I understand what you're hoping to accomplish, but it sounds dangerous.
The threads in Tomcat's servlet thread pool (or any other Tomcat thread pool) should not be tampered with. To assure adherance to the
J2EE spec AND to ensure proper performance and reliability,
you should neither attempt to "capture" and hold one of the Tomcat pool threads nor spawn threads of any kind (pooled or not) as children of any the Tomcat pool threads. Servlet threads should be entered (via call from Tomcat to the servlet's service() method), processed, and returned from as quickly as possible and should not attempt to acquire thread-specific resources (including child threads) or block and wait on long-running operations.
If you want to create your own private thread pool - say for running long-running processes - do that from a ServletContextListener. You can create as many threads (and thread pools) as you like there.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.