I've tried everything I can think of with the most recent 1.7.10 version's config and I cannot find a way of enabling Enchanted String as a usable part as long as Iguana Tweaks is enabled. I've tried whitelisting everything in the main.cfg file, blacklisting with individual parts enabled in the restrictions.cfg file (which honestly maybe I'm getting the material name wrong, but I've tried Enchanted:bowstring, EnchantedFabric:bowstring, TConstruct:bowstring:1 with no usable results)
Seems like it's automatically being blacklisted with no way of whitelisting it. Possibly a bug carried over from the old version? Here's a screenshot of the tooltip while Iguana Tweaks is enabled:

If this is not a bug and just a miscommunication with the config files my apologies, the documentation is self explanatory in most areas but in this area it's kinda fuzzy...

Looks like it's treated like a regular toolpart instead of a custom material one.

Is it something I can work around in the configs? Or is it something I'd need to look into the code for?

It's a code issue.

Hm, this is not really trivial to fix. But you should be able to enable it by allowing stone tools in the config.