Unfortunately when removing individual parts of a fully mixed file you are also going to yank out sounds that occupy the same pan postiion and (when using things like R-Mix) frequency range. Usually the bass and snare are panned center as well so they get affected. With bass it's a little easier to avoid with something like R-Mix because you can leave the lower freqs out of the removal. Snare is a little harder because a lot of the crack will likely be around where your vocals are.
What you can try is automating the "Inside" level in R-Mix so it only removes the target sound as it occurs. In this case automate the fader to only turn down when the there is singing. It will of course still yank out competing frequencies at those points but might be a little better.
To compensate (if you want to get really crazy) you could use synths/samplers/MIDI to match the original parts as best as possible and blend them in to the full mix. That way when R-Mix yanks those sounds out it gets masked a little better by the synth sounds.
eg: Dial in a snare sound that closely matches the snare in the song and program it in to match the original performance. Then when the removal happens you don't lose your snare. Might want to automate the volume of that too so it gets turned up exactly when the removal happens. Drum Replacer would actually be a good tool to do this type of thing.
However... as noted, this type of thing rarely works perfectly. It's just the nature of mixed audio. Once it's mixed down and all smooshed together it's next to impossible to completely isolate. It's like trying to pull the eggs out of a cake after it's been baked. lol