Personally, I think it would be okay for you to use both “bring forth” and “carry forth” instead of bringing. Although I do feel that “bringing” works well already and simplifies the sentence.

Thank you Jessica, Nanren888 and Mark for all your generous help!

Your question is as to whether you could use "bringing forth" or "carrying forth". You can use the former, but not the latter. But it must be stated differently as the problem is with bringing forth and into. " In this narrative, Bianchi becomes a pivotal figure in bringing forth the maturation of an earlier Roman neoclassical style." The preposition into is not used in the correct context and must be omitted. I hope this assists you. Teacher Mark

to bring forth into sounds a little strange, as bring forth - seems like outward rather than into, but not incorrect, it can mean forth - into focus, to the front. . Perhaps, if you like "bring forth" ... "bring forth its maturation".. rather than "bring forth into its maturation".