 Solution

Use both families of bad events in one asymmetric Lovász local lemma application. Give every four-edge event parameter $x_4$ and every six-edge event parameter $x_6$. A path of length $\ell$ shares edges with at most $4\ell {\Delta }^3$ four-edge paths and $6\ell {\Delta }^5$ six-edge paths. It therefore suffices to verifyThese slightly excessive exponents are safe even though they may count the event itself. Choose $x_4=2/k^2$ and $x_6=2/k^3$. For $k\ge C{\Delta }^{5/3}$, the sums of exponents times their parameters are bounded by constants times $C^{-2}{\Delta }^{-1/3}+C^{-3}$, uniformly in $\Delta \ge 1$. Taking $C$ sufficiently large makes each product of factors at least $1/2$, for example using $(1-x{)}^r\ge 1-rx$ and multiplying the two factors. Both displayed inequalities then hold. Positive probability of no bad event gives simultaneous avoidance of both lengths. Keeping two event parameters avoids replacing the rarer six-edge events by the larger four-edge probability.

These are the avoiding short repetitions in an edge coloring bounds, for repetitively colored paths.