I am not personally aware of a single word that describes one who, or that which creates templates, and while I'm not the ultimate authority I think it is fair to say that if you found an obscure word then it wouldn't be recognised by the average native English speaker.
One of your suggested words "templater" sounded very plausible but doesn't appear to be in any official dictionaries. The word does appear in Wiktionary, and although that doesn't count for much it means that at least somebody has used it before.
However if you are looking to name your software, the great thing about English is that you can use common rules to make up words as names and the meaning will be instantly recognised.
- Templatiser (or Templatizer in the US)
...will be widely understood even though they are not (or may not be!) proper words. The fact that Templater has found its way into a wiki and gone unchallenged proves this point.
Some additional names for your bonus 'name the software' question:
- Mr. Template (for fans of Back to the Future)
- Templat-inator (for fans of Phineas and Ferb)
- Simon Templater
- Templater of Doom
- Templatey McTemplateface (every naming competition has to have this)