Provided you're prepared to do a bit more work, just extract the initrd, copy your application into /usr/bin or /usr/local/bin and package it back up. It does mean that every time you upgrade your Tinycore installation you have to repeat the process, but if you're using it in an embedded application, you probably won't be doing it that often.
It's what I did when developing stuff, although I might yet go back and turn my application into an extension when I've got a bit of spare time. After the effort of compiling a newer kernel for a 486sx and re-packaging everything, one extra copy command was trivial.