I think I've found the answer (maybe writing this down has made me think harder about it).
The application I have in mind uses interpolation, so $K\in \mathbb{N}$ was really just for simplicity's sake, and I can assume $K$ to be odd, $K+1=2L$, $L\in \mathbb{N}$.
Continuing from the end of my attempt: $$ \begin{align} &\int_{\partial M} \nabla_{a_i}w_{na_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} \\ &= \int_{\partial M} \nabla_{a_i}{(\iota_\nu w)}_{a_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} - \int_{\partial M} {(\iota_{\nabla_{a_i}\nu} w)}_{a_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} \\ \end{align} $$ Here the first term vanishes byas integration by parts$\iota_\nu w=0$, while the second term spits out some Christoffel symbols. Since the manifold is compact, we get $O( \int_{\partial M} |w|^{2K+2})$.
We can control this error by the negative quantity (which I previously threw away, prematurely) $$ - \ll d(|w|^2), d(|w|^{2K}) \gg_M = -K \int_M \left|d(|w|^2)\right|^2 |w|^{2(K-1)} $$ Let $f=|w|^2$, then we want $$C\int_{\partial M} f^{K+1} - \int_M |df|^2 f^{K-1} \lesssim \int_M f^{K+1}$$ where $C$ is some constant (depending on $k,K,M$). As $K+1=2L$, set $F=f^L$, and our problem, after simplifyingsimplification, becomes $$C\int_{\partial M} F^2 - \int_M |dF|^2 \lesssim \int_M F^2$$
This is just Ehrling's inequality. As $\tau:H^1(M)\to L^2(\partial M)$ compact and $H^1(M)\hookrightarrow L^2(M)$ continuous, we have $$ \forall \epsilon >0, \exists C_\epsilon > 0: ||\tau F||_{L^2(\partial M)} \leq \epsilon ||F||_{H^1} + C_\epsilon ||F||_{L^2} $$
I am very excited that this turns out to be true. This further confirms my belief that the above setting is, in a sense, universal.