The distinction between register and instrumentation is that register is bound to pitch and underlies instrumentation. Every instrument and vocal usually breaks into 3 registers that can be classified in 2 general types: intensity growing towards the top (vocals and brass) or 5towards the bottom (reed woodwinds and strings). This typology goes against and across the distinctions between different timbres of the instruments. However, the distinctions between registers can be greatly reduced. For instance, the bel canto training can completely conceal a breaking point between neighboring registers. Also, the timbral differences between different instruments (and vocals) greatly exceed the timbral differences between different registers of the same instrument (or voice type).Yet another important distinction is that register plays a formative role for tonal organization of modes of timbre-oriented music that are characterized by indefinite pitch (relative and variable pitch values), such as ekmelic and khasmatonal modes. In such modes, the degrees are defined in regards to their position within a vocal register(s). It is possible that the same principles are in play in the instrumental forms of music of the same ethnicities that keep cultivating such vocal music (e.g., music for Jaw Harp or musical bow). The aspect of instrumentation completely misses this formative melodic modal function. Combinations of timbral colors of different instruments do notform specific musical modes. Timbral coloration is known to be modally formative only in instrumental ensembles consisting of the sameinstrumental types -e.g., a set of gongs. But then, such cases fall within the domain of register rather than instrumentation.On the other hand, instrumental timbres often blend, forming new composite colors (for example, clarinet + oboe). There is nothing remotely similar in the domain of register -registers don't blend.It can be generalized that register fundamentally opposes instrumentation: register is based on timbral similarity, whereas instrumentation -on timbral contrasts. Composers select a specific instrument to "color" constituent sounds in a musical composition in different colors. Singers (and possibly instrumental players) usually select a specific register to secure unityin timbral coloration for the pitch-classes of a musical mode. Timbral contrasts are important in khasmatonal music, where a mode is defined by the group of pitch classes of one register contrasting the other register (e.g., falsetto or rasping). However, such music is rather rare and is still operated by the principle of integratingtimbral colors into a melodic phrase rather than by differentiation of timbral color to color the music textures, as it occurs in instrumentation. The only exception is Schoenberg's experiments with Klangfarbenmelodie that did not work -and could not work, because changes of instrumental timbres within the same melodic stream has been demonstrated to segregate this stream into fragments that obstruct phrasing.

If you are interested, here is a thorough critical overview of Riemannian approach:Rehding, Alexander. 2003. Hugo Riemann and the Birth of Modern Musical Thought. Cambridge University Press.Below are a couple of reviews of the applicability of Riemannian principles to jazz:McGowan, James. 2010. “Riemann’s Functional Framework for Extended Jazz Harmony.” Intégral 24: 115–33. https://www.jstor.org/stable/41495296.Capuzzo, Guy Capuzzo. 2004. “Neo-Riemannian Theory and the Analysis of Pop-Rock Music.” Music Theory Spectrum 26 (2): 177–200. https://doi.org/10.1525/mts.2004.26.2.177.All Soviet and Russian musicology is based on Riemann