An exponentially-ambiguous weighted automaton without an equivalent polynomially-ambiguous automaton
Here is a natural candidate: the alphabet is $\{a,b,c\}$, and to a word $v=u_1cu_2c\dots cu_k$ with each $u_i\in\{a,b\}^*$, you want to associate $$f(v)=\sum_{i=1}^k \min(|u_i|_a,|u_i|_b)$$

There is a straigthforward exponentially ambiguous automaton for this $f$: each time you see a $c$, guess if you go to a state counting $a$'s or to a state counting $b$'s.
It should not be too hard to prove that only exponentially ambigous automata recognize this function $f$. As a very sketchy argument, consider the fact that between two $c$'s, you must either count the number of $a$ or the number of $b$, so you need two different states to do so. Since you must be able to oscillate between these states, it means that you have distinct paths of the form $p\stackrel{w}{\longrightarrow} p$ for some state $p$ and word $w$ (containing some $c$'s).