= Solution
The number of <spaxels> is $M=(\beta/\theta)^2$. To obtain definite scaling exponents, keep the <spectral resolving power>, <wavelength> interval, <diffraction grating> <angles> and groove spacing, input <focal ratio>, and <photodetector> sampling of a <spectral resolution element> fixed. A fixed <photodetector> then accommodates a fixed number $M_0$ of spectra, so $N\simeq M/M_0$ (rounded up in an actual instrument). The corresponding <etendue> per <spaxel> scales as $\theta^2$. Angular ratios are independent of whether both <angles> are in arcseconds; optical invariant equations use radians.
Write $b$ for the collimated beam <diameter>. The <diffraction grating> result $R\theta D=W(\sin\alpha+\sin\beta)$ gives $b\propto W\propto\theta$. A fixed <collimator> <focal ratio> gives $f_{\rm coll}\propto b\propto\theta$. With a fixed image width $p$ in <photodetector> <detector pixels>, the invariant $\theta D=pA_{\rm cam}/f_{\rm cam}$ and $A_{\rm cam}\propto\theta$ instead imply $f_{\rm cam}$ is constant: the <optical camera> <focal ratio> increases as $1/\theta$.
In a simple on-axis beam-envelope model, each <collimator> and collimated disperser space has area proportional to $b^2$ and length proportional to $b$, whereas the <optical camera> cone has area proportional to $b^2$ and fixed length. Hence
$$
\boxed{V_{\rm coll}\propto\theta^3,\quad V_{\rm grat}\propto\theta^3,\quad V_{\rm cam}\propto\theta^2,\quad
V_{\rm total}\simeq M(a\theta^3+b_0\theta^2).}
$$
The constants include $1/M_0$ and fixed design parameters. At fixed field size, the combined <collimator> and <diffraction grating> volumes scale as $M\theta^3\propto\theta\propto M^{-1/2}$, while the combined <optical camera> beam-cone volume scales as $M\theta^2\propto1$.
These are <integral-field spectrograph volume scaling> laws for the shrinking beam, not a claim that the whole apparatus can shrink without a floor. A <photodetector> of fixed transverse size $d_0$ needs space for its field: an <optical camera> envelope interpolating between <optical pupil> width $b$ and <photodetector> width $d_0$ has volume proportional to $f_{\rm cam}(b^2+bd_0+d_0^2)$, rather than just $f_{\rm cam}b^2$. Replication then adds terms scaling as $M\theta$ and $M$, which can grow as the <spaxels> shrink. Other field clearances and mechanical margins also change the asymptote. Without the fixed spectral/design assumptions above, the information in the question does not determine unique physical-volume exponents. The role of fixed <photodetector> size and beam spread is discussed in https://academic.oup.com/mnras/article/376/3/1099/1746825[Allington-Smith's instrument scaling model].
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