Firstly it's nice to see someone making use of RS5000! Sadly, though, it works in the same way as most samplers in that it achieves the pitch changes simply by playing back the sample data at different speeds.
Within Reaper you could do the work using ReaPitch which would also allow you to do formant correction to compensate for the vocal sounding unrealistic (that 'chipmunk' effect) when pitched. The process would be quite a bit slower than simply dropping the sample into a sampler and playing the pitches though.
In order to do what you're asking with a sampler, the instrument must be able to timestretch the sample data. This is usually achieved using granular techniques where the sample is cut into tiny slices which can be compressed together or spread apart and smudged somewhat in time to compensate for the playback speed - and therefore keeping the sample length the same. There are usually audio artefacts associated with granular playback which mean the effects of the stretching can be fairly obvious - especially on harmonically complex material such as vocals. With some tweaking of the various parameters it's usually possible to minimise these somewhat but using vocal samples the end product is never going to be something you'd consider realistic.
Some examples of sample playback capable instruments which have this ability are:
- TAL Sampler
- Native Instruments Absynth
- Camel Audio Alchemy
- Ableton Sampler (possibly Simpler also but I'm not 100% sure)
Good results can also be achieved using additive resynthesis where the audio is analysed and rendered using a set of sine waves. This opens up a lot of creative possibilities for bending and mangling the audio in ways not possible with samples.
Camel Audio Alchemy and Image Line Harmor are a couple of examples of synths which can carry out additive resynthesis.