Almgren (https://mathscinet.ams.org/mathscinet-getitem?mr=855173) proves that the Plateau solution will satisfy the 2-dimensional Euclidean isoperimetric inequality. So you can bound
$$
\textrm{Area(Plateau Solution)} \leq (4\pi)^{-1}(L(\gamma_1)+L(\gamma_2)+L(I_T))^2.
$$
If you just want a coarse estimate you should be able to bound each term by the $L^\infty$ norm of $b_1,b_2$.

Of course, this inequality is unlikely to be close to optimal. (For example, if $b_1=b_2$ then the area of the Plateau solution is $=0$ while the RHS above is nonzero). If you want an estimate that works better in this case you might construct the explicit competitor
$$
\Gamma : = \bigcup_{t=0}^T I_{\Phi_1(t,x_0),\Phi_2(t,x_1)} \times \{t\}
$$
namely you union up all the lines between the points in the flow. You should be able to compute the area of this in terms of $b_1,b_2$ using the co-area formula.

EDIT: As requested here is a formula for the area of $\Gamma$. I did it via a parametrization instead of co-area but I think co-area would also work.

I parametrize $\Gamma$ by $F: [0,1]\times [0,T]\to\mathbb{R}^{n+1}$
$$
F(s,t) : = (s\Phi_1(t)+(1-s)\Phi_2(t),t)
$$
(dropping the $x_0$ notation). Then we have
$$
\partial_s F = \Phi_1-\Phi_2, \qquad \partial_t F = s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2 + \mathbf{e}_{n+1}.
$$
In particular, the induced metric is
\begin{align*}
g_{ss} & = |\Phi_1-\Phi_2|^2 \\
g_{st} & = \langle \Phi_1-\Phi_2,s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2\rangle\\
g_{tt} & = 1 + |s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2|^2.
\end{align*}
Hence
$$
\det g = (1 + |s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2|^2)|\Phi_1-\Phi_2|^2 - \langle \Phi_1-\Phi_2,s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2\rangle^2.
$$
This gives the explicit formula
$$
\textrm{area}(\Gamma) = \int_{[0,1]\times [0,T]} \sqrt{(1 + |s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2|^2)|\Phi_1-\Phi_2|^2 - \langle \Phi_1-\Phi_2,s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2\rangle^2}.
$$
If you want to write this in terms of $b_1$ and $b_2$ you can do the following (perhaps one can be more precise here to get a better estimate). First of all
$$
|\partial_t \Phi_i| = |b_i| \leq \Vert b_i\Vert_{C^0}
$$
Secondly, by the argument here we can bound
$$
|\Phi_1-\Phi_2| \leq L^{-1}(e^{Lt}-1)|b_1-b_2|_{L^\infty}.
$$
for $L$ the Lipschitz constant of one of the vector fields.

Using this above we should get
$$
\textrm{area}(\Gamma) \leq (1 + (|b_1|_{C^0}+|b_2|_{C^0}) |b_1-b_2|_{C^0}^2 L^{-2}(e^{LT}-LT-1).
$$

Note that I discarded the $g_{st}$ term, which could improve this estimate if you had some way of knowing it was big. But it does have the nice feature that it vanishes if $b_1=b_2$ and it's $O(T^2)$ for $T$ small.

I'm sure you can improve the estimate used above in the case of large $T$ (e.g. if $b_i$ are bounded, then the flows should diverge at most linearly).