Is it permissible to continue a written conversation with a committed Muslim girl, within the mentioned guidelines, with her mother's knowledge, and without the potential for fitna given the age and cultural differences?

It is not permissible to correspond with a non-mahram woman via the internet or any other means, because it is a gateway to fitna (temptation/strife) and a means to corruption. Even if her mother knows, or if the woman is older, or if the conversation is about general and polite matters, none of this justifies the continuation of the relationship, as it could be a gradual enticement from Satan. Speaking with a young non-mahram woman is not permissible without a need, because it is a cause for fitna. Therefore, one must fear Allah and sever this relationship.