Measuring a high Michaelis constant using fluorescence
Based on your description, it sounds like you are measuring the production of hydrogen peroxide by an oxidase enzyme. In that case, it is not necessary for the Amplex Red to be at the same concentration as the substrate. You only need enough reagent to determine the initial rate of reaction at each substrate concentration, ie when only a small portion of the substrate has been consumed.
If $\pu{5 μM}$ is not sufficient for that purpose, you can determine the resorufin (produced by reaction of Amplex Red with $\ce{H2O2})$ by absorbance rather than fluorescence for lower sensitivity and higher maximum. According to the Thermo Fisher assay kit info, resorufin has an extinction coefficient of $\pu{58000 M^-1 cm^-1},$ so with a standard $\pu{1 cm}$ path length cuvette, your linear range would probably go up to around $\pu{50 μM}$ depending on your spectrophotometer. With a short path cuvette (these are available with $\pu{0.2 cm}$ and $\pu{0.1 cm}$ path lengths), you could go quite a bit higher, certainly more than sufficient for your needs.
As for why you cannot just scale the concentrations down, consider the limiting case where the Michaelis constant is equal to the $K_\mathrm{d}$ of the substrate. We have that
$$K_\mathrm{d} = \frac{[\ce{E}][\ce{S}]}{[\ce{ES}]},$$
so $[\ce{S}] \approx K_\mathrm{d}$ when $[\ce{E}] = [\ce{ES}]$ (assuming $[\ce{S}]_\mathrm{tot} \gg [\ce{ES}]).$ Thus, the absolute concentration of $\ce{E}$ is not a factor, as long as it is small enough that the amount of $\ce{S}$ bound does not substantially change the concentration of free $\ce{S}.$ You should get the same result for $K_\mathrm{M}$ regardless of what concentration of enzyme you use.