The second part of the conjunction is adding more to waht is stated. Yes, not knowing whom is being served is rewarded, but not necessarily believing they were serving another person by mistake. The faith alone is a discerning element between the two scenarios.
I also agree that there might be some who do not know Christ by a particular name, yet know Him through faith, as He is also known by over some 350 names in Scripture.
I suspect there are many Muslims who might actually be saved, but would be reclassified as backslidden believers focusing on the soulish rather than returning to Him by faith in Him.
It is interesting to observe how many other religions attempt to counterfeit or synthesize the experience of the spirit by physical and soulish behavior. Acts that come to mind are whirling dirvishes, cutting the forehead in Islam, chanting mantras, and many of the figures that are depicted in other religious symbols that are from spiritual experience. A major difference between those religions and fellowship with God is that men attempt to bring God to them in those religions, while through Christ, we already have the Word available to us and indwelling in us while in fellowship with Him, yet experience isn't mandated, rather continuing sanctification is the immediate direct benefit.