Certainly not all the members of the Empire Base Ball Club were members of the StLFD. Al Spink wrote in The National Game that the club "had in its ranks many wide-awake business men as well as some of the most influential mechanics and tradesmen. It had for its officers the most popular men in the community-men selected for their great heart, wide acquaintance and numerous following." The Empires were by no means an extension of the StLFD but the fire department had both a strong presence and influence on the club.
This is not unique in the history of 19th century baseball. According to Warren Goldstein in A History of Early Baseball, one of the "most fertile sources of baseball nines were volunteer fire companies..." The most famous example of this was the New York Mutuals who were "founded in 1857 by the Mutual Hook and Ladder Company No. 1." It's Goldstein's contention that the volunteer fire companies and the early fire departments played a vital role in the development of early baseball, "providing a cultural bridge between this new sport and the earlier, more rough-and-tumble world of working-class leisure." He goes on to list some of the similarities between the two institutions including their names, social activities, and uniforms.
It's doesn't appear, based on Spink's observations on the make-up of the club, that the Empires fit Goldstein's pattern exactly. But, under the leadership of Sexton, a relationship between the club and the StLFD was established and this relationship was used to the advantage of the club. This can be seen in Tom Oran's switch from the Union Club to the Empires. Peter Morris, in his essay on Oran for SABR's Biography Project, writes that "(on) June 5, 1869, the Empire Club defeated the Unions to regain local supremacy. Shortly afterward, the Empires lost their catcher to injuries and recruited Oran to take his place...Both clubs appear to have been amateurs, and it is unlikely that Oran was offered money to change clubs. It is, however, quite possible that he received another sort of inducement to join the Empires. Empire club president Henry Clay Sexton was the chief of the St. Louis fire department and Oran was soon working as a city fireman."
I think it's safe to assume that there were more members of the Empire Club who were also members of the StLFD than the five that I'm aware of. I'm currently searching for a list of members of the StLFD in the 19th century in order to compare it to known members of the Empire Club. When these two lists are cross-checked then the extent of the relationship between the two organizations should become clearer.
Note: The picture at the top of the post is of a funeral procession of a St. Louis fireman who lost his life in the line of duty in 1916. It's the earliest photograph of St. Louis firemen that I've been able to find and was taken from History's Time Portal to Old St. Louis.