No such $M$ exists for the following $\sigma$. Partition $[0,1]$ into countably many intervals, with endpoints $t_0=0,t_1,t_2,...$ and let $\sigma$ take value 1 on the odd ones and value 2 on the even ones. Given any $M$, the following event $A_M$ has positive probability:
$A_M$ requires that the increments $W_{t_k}-W_{t_{k-1}}$ are in $(9,10)$ for the first $M$ odd values of $k$, and in $(-6,-7)$ for the first $M$ even values of $k$, with $W_t-W_{t_{k-1}}>-1$ for $t \in [t_{k-1},t_k]$ when $0<k<2M$.
Then given $A_M$, we have $W_{t_{2M}}>2M$, so the event $W_1>M$ holds with high probability, yet on $A_M$, the stochastic integral considered is $<-2M$ if you integrate up to $t_{2M}$, and the integral over $[t_{2M},1]$ is likely to be $<2M$.
Let $ B $ be a Brownian Motion, $\mu_{B}=\mathsf{P}\circ B^{-1} $ and $ Z $ be a continuous local martingale, $ \mu_{Z}=\mathsf{P}\circ Z^{-1}$.
Then $ \mu_{Z}\ll \mu_{B} $ if and only if $ \mu_{Z}=\mu_{B} $.
In this case, if $ Z $ may be expressed as Ito stochastic integral of a predicatble $ \phi $ with respect to $ B $:
\begin{equation*}
Z(t,\omega)=\int_0^t \phi(s,\omega)\,\mathrm{d}B(s,\omega)
\end{equation*}
Then processes $ |\phi|=(|\phi(s,\omega)|,s\ge 0) $ and 1 are indistinguishable, i.e., for almost all $ \omega $ trajectories $ \phi(\cdot,\omega)\equiv 1 $.
Now we prove above facts. Since $ Z $ and $ B $ are continuous local martingale, for its predictable variation process $ \langle Z,Z\rangle$ and $\langle M,M\rangle $ the following facts hold, (cf. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Corrected 3rd Ed. Springer, 2005, p124. Th.4.1.8.)
\begin{gather*}
\textrm{pr}-\lim_{n\to\infty}\sum_{k=1}^{2^n}\Big[Z\Big(\frac{kt}{2^n}\Big)- Z\Big(\frac{(k-1)t}{2^n}\Big)\Big]^2=\langle Z,Z\rangle_t,\qquad \forall t>0.\tag{1}\\
\textrm{pr}-\lim_{n\to\infty}\sum_{k=1}^{2^n}\Big[B\Big(\frac{kt}{2^n}\Big)- B\Big(\frac{(k-1)t}{2^n}\Big)\Big]^2=t,\qquad \forall t>0. \tag{2}
\end{gather*}
(2) is equivalent to the following,
\begin{equation*} \lim_{n\to\infty}\mu_{B}\Big(\Big|\sum_{k=1}^{2^n}\Big[x\Big(\frac{kt}{2^n}\Big)- x\Big(\frac{(k-1)t}{2^n}\Big)\Big]^2 - t \Big|>\epsilon\Big)=0,\quad \forall \epsilon >0,\quad \forall t>0. \tag{3}
\end{equation*}
Due to $ \mu_{Z}\ll \mu_{B} $ and (3), as $ n\to\infty $,
\begin{align*}
&\mathsf{P}\Big(\Big|\sum_{k=1}^{2^n}\Big[Z\Big(\frac{kt}{2^n}\Big)- Z\Big(\frac{(k-1)t}{2^n}\Big)\Big]^2-t \Big|>\epsilon\Big)\\
&\quad =\mu_{Z}\Big(\Big|\sum_{k=1}^{2^n}\Big[x\Big(\frac{kt}{2^n}\Big)- x\Big(\frac{(k-1)t}{2^n}\Big)\Big]^2 - t \Big|>\epsilon\Big)\\
& \quad \longrightarrow 0,\quad \forall \epsilon >0,\quad \forall t>0. \tag{4}
\end{align*}
comparing (1) and (4), get
\begin{gather*}
\mathsf{P}(\langle Z,Z\rangle_t=t)=1, \quad\forall t>0,\\
\mathsf{P}(\langle Z,Z\rangle_t=t, \forall t>0 )=1 \tag{5}
\end{gather*}
Now from the Lévy's characterization of Brownian Motion, $Z$ is a Brownian motion and
$\mu_Z=\mu_{B} $.
Remark: The conclusion in following book is also useful: C. Dellacherie & P. Meyer, Probabilities and Potential B, volume 72 of North-Holland Mathematics Studies.
North-Holland, Amsterdam, 1982.
Best Answer
Fix a finite interval $I$.It suffices to show that almost surely, $$\mathcal L(\{t \in I \, | \, B_t = A_t \}) = 0 \,.$$ Brownian motion restricted to $\{t \in I \, | \, B_t = A_t \}$ has bounded variation, so a positive answer is implied by the following stronger result, a special case of Theorem 1.3 of [1]:
Theorem[1] Let $\{B(t): t\in [0,1]\}$ be a standard Brownian motion. Then, almost surely, for all $S\subset [0,1]$, if $B|_{S}$ is of bounded variation, then $\overline{\dim}_M S\leq 1/2$.
Here $\overline{\dim}_M$ is the upper Minkowski dimension (a.k.a. upper box dimension.)
[1] Angel, Omer, Richárd Balka, András Máthé, and Yuval Peres. "Restrictions of Hölder continuous functions." Transactions of the American Mathematical Society 370, no. 6 (2018): 4223-4247. https://www.ams.org/journals/tran/2018-370-06/S0002-9947-2018-07126-4/S0002-9947-2018-07126-4.pdf http://wrap.warwick.ac.uk/84253/7/WRAP-restrictions-continuous-functions-Peres-2018.pdf