Although the first Chinese character in this term, 番, could mean foreign, the term in the question, 番菜馆, referred to restaurants which served Western food. There were renowned examples under this category, for instance, 一品香番菜馆, which literally means First Grade of Fragrance Western Restaurant, and 撷英番菜馆, which literally means Collecting the Outstanding Western Restaurant. The former was in Shanghai, whereas the latter was in Beijing.
And you could find out that back then, lots of well-known people patronized this kind of restaurant. For instance, in 1933, when a writer Shen Congwen, or 沈从文 (1902–1988) in Chinese, got married with Zhang Zhaohe, or 张兆和 (1910–2003) in Chinese, they held a sumptuous wedding banquet in Collecting the Outstanding Western Restaurant in Beijing.
A scholar Ji Xianlin, or 季羡林 (1911–2009) in Chinese, was invited to this occasion. And he wrote down this in his diary and mentioned that many luminaries attended this banquet. Here is the excerpt in Chinese in a book, Diary of Ji Xianlin, or 季羡林日记 in Chinese: 他同张兆和女士结婚, 在北京前门外大栅栏撷英番菜馆设盛大宴席, 我居然也被邀请. 当时出席的名流如云.
Here is a picture of Shen Congwen and Zhang Zhaohe.