What is the ruling on doing something that outwardly resembles an act of the innovators (Mubtadi’ah), even though the doer, being from Ahl al-Sunnah, does not believe it to be a Sunnah, while some people do believe it to be a Sunnah? And is this considered an agreement with them in their belief?

Your question is unclear, and we do not know whom you mean by "innovators." A person should not accuse everyone who does something they imagine to be an innovation of being an innovator. Applying perfume for prayer is legislated due to the great benefits it entails, such as following the Sunnah of the Prophet (peace and blessings be upon him), adorning oneself, and bringing joy to oneself and the Muslims. It is a good deed for which one is rewarded with a righteous intention. Your family's belief that applying perfume is only for prayer is a mistaken belief, but they are not called innovators for it. Rather, they should be advised and informed that the Prophet (peace and blessings be upon him) loved perfume and that it is legislated for a Muslim in all circumstances, and its desirability is even more emphasized in public gatherings. Ibn Qudamah said: "It is desirable to apply perfume because the Messenger of Allah (peace and blessings be upon him) loved perfume and used to apply it frequently."