Some notes in Taiwanese cities as this was quite an interesting time in Taiwanese history. Chinese settlement of Taiwan was pretty haphazard during the early-middle Qing when the entirety of Taiwan was governed as part of Fujian province, and a lot of place names involved Taiwanese Minnan and Hakka transliteration of indigenous names from the tribes in the area. Larger changes occurred towards the late 1870s-1880s when it was upgraded to a province outright and started having walled administrative cities known as fu.
Taipei didn’t exist as a municipality until late in the Qing period, like 1880s. It was previously a collection of small riverside settlements like Tamsui, Bangka (Wanhua), and Twatiutia (Dadaocheng) which didn’t really get going and merge until the late Qing and Japanese period.
New Taipei literally didn’t exist as an entity until 2010, when Taipei County (collection of towns making up the suburbs of Taipei) was reorganized into a special municipality. Taipei County didn’t exist until the ROC period post 1949.
the specific spelling and characters of Keelung would have been slightly different as something more similar to Kelang/Ke-lang, which itself was a Minnan transliteration of an indigenous word. The characters 基隆 would have been conferred in the late 1800s provincial upgrade.
Taoyuan (Tho-a-hng in Taiwanese Minnan) may have existed as a local name but it certainly wasn’t significant enough to be noted as a city. It only really became a city of note in the Japanese era.
Hsinchu was known as Zhuqian/Tek-kham and only became Hsinchu (lit. New Zhu(bamboo)) during late Qing reorganization in the 1880s.
interestingly Chiayi (Minnan: Kagi) was conferred as a name by the Qing emperor just a few decades before 1802 as a celebration of squashing the Lin Shuangwen rebellion. Prior to that it would have been known as Tirosen/Tsulosan, again a transliterated indigenous word.
Tainan as a name likely didn’t exist during this time, being known rather as Taiwan-fu or Anping
The name Kaohsiung could not have existed in 1802, as the current name comes as a post 1949 Mandarin graphic reading (高雄 Gaoxiong) of a Japanese transliteration (高雄 Takao) of a Minnan transliteration (打狗 Ta-kau) of a indigenous word (takao).