Was Ibn Hajar al-Asqalani truly a Sufi, and to what extent is his affiliation with one of the Sufi orders accurate, based on what was mentioned in the introduction to Fath al-Bari and other sources?

One should differentiate between the types of Sufism. Some are praiseworthy, such as the Sufism that aims for asceticism and detachment from worldly life, and some are blameworthy, which is the Sufism that includes innovations (bid'ah) and polytheistic practices (shirk).

Al-Hafiz Ibn Hajar loved the righteous and detested innovators and misguided individuals. This is evidenced by his denouncement of Ibn Arabi and some of the objectionable practices of Sufi orders.

The general principle is to shun the people of innovation, but this shunning is governed by religious benefit (maslaha shar'iyya). Ibn Hajar may have taught in some schools where Sufism was present in order to achieve a greater benefit. His dissemination of Hadith and Sunnah and his teaching of knowledge to people were a form of da'wah (invitation to Islam) and a way to achieve a greater good than abandoning those schools.

There are two scholars named "Ibn Hajar": Al-Hafiz Ibn Hajar al-Asqalani, and Ibn Hajar al-Haytami, who came later than him. The one usually associated with belonging to Sufi orders is Ibn Hajar al-Haytami, not al-Asqalani.