What purpose did "道搿" serve?
The fact that it is a transliteration of English "dog" points to its possible use and appearance. Perhaps, it was used for those foreign breeds of dogs that looked different from the breeds that Chinese people were accustomed to at that time (end of Qing dynasty).

But also note that this particular word (道搿) does not seem to appear in any other text than the mentioned novel (文明小史). So it might have been simply the novelist's invention, that the authors of Hanyu Da Cidian still felt they needed to explain. (Mind you that this dictionary was compiled before widespread use of corpus linguistics, so the authors wouldn't really know if this word is just a one-time appearance across all of the Chinese literature, or if it got some traction and started to be used by others.)