Remove custom template from Word 2003

I think you should be able to remove the no-longer-in-existence template by unloading it. From Load or unload a global template or add-in program on Office's help website:

Unload a global template or add-in

• On the Tools menu, click Templates and Add-Ins.
• Click the Templates tab.
• To unload a template or add-in but leave it in the Global templates and add-ins box, clear the check box next to the name of the item.
• To unload a template or add-in and remove it from the box under Global templates and add-ins, click the item in the box and then click Remove.

and from About loading and unloading global templates and add-in programs:

To conserve memory and increase the speed of Word, it's a good idea to unload templates and add-in programs you don't often use. When you unload a template or add-in that's located in your Startup folder, Word unloads the template for the current Word session but automatically reloads it the next time you start Word. When you unload a template or add-in located in any other folder, it is unavailable until you reload it. To delete a template or add-in from Word, you must remove the template or add-in from the Templates and Add-ins dialog box.