Magnetars, isolated neutron stars with ultra-strong magnetic fields (10^14--10^15\,G), exhibit complex spin-down behavior that cannot be described by canonical magnetic dipole braking (n=3) due to strong timing noise and glitches, which prevent direct measurement of the instantaneous braking index. To circumvent this, we employ the equivalent average braking index \langle n \rangle inferred from supernova remnant (SNR) ages for a sample of eight associated magnetars, yielding values ranging from \sim 1 to 42, indicating multiple competing energy-loss channels. We develop a purely kinematic, model-independent framework to translate these \langle n \rangle values into quantitative constraints on the time-averaged torque fractions of four channels: particle wind (k=1), magnetic dipole (k=3), mass quadrupole GW (k=5), and r-mode current quadrupole GW (k=7). The key advance is a weighted-average relation connecting \langle n \rangle to the mean torque fractions \barf_i: \langle n \rangle - 3 = -2\barf_1 + 2\barf_5 + 4\barf_7, with the normalization \sum \barf_i = 1. This allows us to analytically derive strict lower and upper bounds for the r-mode torque fraction \barf_7. The lower bound, \barf_7,\min = \max(0, (\langle n \rangle - 5)/2), arises from assuming the highest non-r-mode exponent (mass quadrupole) dominates the competing channels, while the upper bound, \barf_7,\max = (\langle n \rangle - (\delta + 3\varepsilon + 5\gamma))/7, incorporates physically motivated minimum contributions from other channels, where \delta,\varepsilon,\gamma denote the minimal torque fractions for wind, dipole, and quadrupole, respectively. We also incorporate a correction for magnetic field decay, demonstrating that failing to subtract it would artificially inflate \barf_7 for sources with \langle n \rangle \gg 7, where the linear programming framework becomes inapplicable. Applying the framework to the eight sources, we find that SGR~0501+4516 (\langle n \rangle = 6.3) requires a significant r-mode contribution (\barf_7,\min \gtrsim 0.65), while low-\langle n \rangle sources permit \barf_7 \simeq 0, indicating r-modes are not mandatory. Concurrently, we estimate the GW strains from both channels using standard quadrupole formulae with fiducial parameters (ellipticity 10^-6, r-mode amplitude 10^-5, distance 10~kpc): typical magnetars produce strains h_0 \sim 10^-26 to 10^-24 at sub-Hertz frequencies (0.17--1~Hz), far below the sensitivity of current (aLIGO) and near-future (Einstein Telescope) detectors, rendering direct detection unfeasible in the near term. Despite this, the derived upper limits on \barf_7, combined with non-detection constraints from LVK O4a and multi-wavelength observations (e.g., IXPE polarimetry, NICER timing), can effectively compress the allowed parameter space for the r-mode saturation amplitude \alpha and the magnetic ellipticity \epsilon_B, narrowing them below pure theoretical estimates. This work provides a robust, observation-driven pathway to constrain magnetar GW emission without relying on direct detection, bridging electromagnetic astronomy and GW astrophysics and offering a practical tool applicable to existing and future magnetar samples in the multi-messenger era.