Local time of Brownian motion + Lipschitz continuous function Let $\mathrm{ Lip} (M)$ denote the space of all functions on $[0,T]$ with Lipschitz constant and $L^\infty$ norm bounded by $M$. Let $(B_t)_t$ be a Brownian motion defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Does the following lemma hold?
Lemma: For all $M>0$ and almost all $\omega\in\Omega$, there is a constant $C=C(M,\omega)$, such that for all $\epsilon>0$,
$$\sup_{g\in \mathrm{ Lip}(M)} \int_0^T 1_{|B_t+g(t)|<\epsilon}dt<C\epsilon.$$
 A: From scaling relations, we may assume that $T=M=1$.
**** The argument I sketch below only gives a weaker result than in the OP, so the question is still open. Thanks to Nina for pointing this out. ****
Step 1: Introduce the random variable 
$$Z_\epsilon=\sup_x \int_0^\epsilon 1_{|B_t-x|<2\epsilon} dt
\leq \sup_{j} \int_0^\epsilon 1_{|B_t-j\epsilon|<4\epsilon}dt, $$
and set $M_\epsilon=Z_\epsilon/\epsilon^{3/2}$. Using the heat kernel and the second inequality above, we obtain that $EM_\epsilon\leq C_1$ and
$EM_\epsilon^2\leq C_2$. (This requires a detailed computation, which I am skipping here, so this needs to be double checked. There may be a log correction.)
Step 2: Let $M_\epsilon^{(i)}$ be independent copies of $M_\epsilon$. By conditioning on the Brownian motion at times $(i-1)\epsilon$ and forgetting about the Lipschitz condition between the intervals, you get that the random variable in the OP is stochastically dominated by 
$$\epsilon^{1/2} \cdot \epsilon \sum_{i=1}^{1/\epsilon} M_\epsilon^{(i)}=: \epsilon^{1/2} S_{\epsilon}\,.$$
Step 3: We have that $ES_\epsilon=C_1$ and therefore, by Markov's inequality and the estimate on $EM_\epsilon^2$,
$$P(S_\epsilon>2C_1)\leq P(S_\epsilon-ES_\epsilon\geq C_1)\leq \epsilon \frac{C_2}{C_1^2}\,.$$
Step 4: By interpolation, it is enough to consider the sequence $\epsilon_j=2^{-j}$ and apply Borel-Cantelli to conclude the estimate $\epsilon^{1/2}$ in the right hand side of the OP.
A: I first attempted to make this a comment, but lacked the reputation for it. The following is a cryptic sketch which you should consider:
Choose some $g$ which is $Lip(M)$ and define
$$W_t := B_t + g(t).$$
Under an explicit change of measure, i.e. under Girsanov, $W_t$ becomes a Brownian motion.
We can therefore find a specific $C$ for each $g,$ and we know how to bound the moments of $C$ using the explicit change of measure. (Since the local time of a BM is distributed as its maximum below zero), and these will be computed using the measure change, so the moment bounds will be in terms of $M.$
Do this for a dense set of $g$ and pass to a limit?
