Let $W$ be a Coxeter Group with generators $s_1, \cdots, s_n$ and $l:W \to \mathbb N$ be the lenght function, and $T \subset W$ the subgroup of reflections. Then $$l(w)<l(tw) \iff l(w)<l(wt)$$ for every $w \in W$, $t \in T$.
Thoughts: I have tried using that $l(x)= l(x^{-1 })$ and that we can write $t$ explicitally i.e., if $w=s_1s_2\cdots s_k$ is a reduced form for $w$ and $l(tw)<l(w)$ then $t= s_1s_2\cdots s_i\cdots s_2s_1$ for some unique $i$.
If I'm not mistaken, you are not able to prove the claim, since it is not true.
Consider the Coxeter group $\tilde A_1=I_2(\infty)$ with the Coxeter diagram $\circ\underset{\infty}{--}\circ$ and generators $S=\{\,s,t\,\}$. Let's take the element $w=st$, which has $\ell(w)=2$. But $tw=tst$ has $\ell(tw)=3$, where $\ell(wt)=\ell(s)=1$.
Therefore we found $w\in W, t\in T$ (even $t\in S$) with $$\ell(w)<\ell(tw) ~\text{ but }~ \ell(w)>\ell(wt)$$
Where did you get the claim from? Is there a restriction to the Coxeter group? Or do you want to use a length function with respect to some other generating set?
In retrospect, I don't know why I used $\tilde{A}_1$. This should work with every Coxeter group.