Find out if a path exists avoiding circular obstacles
Your assumption using binary search is a good assumption, that means
that we can assume we know the radius, and focus only on the subproblem:

Given a rectangle and a set of circles, is it possible to navigate in
the rectangle from the top left corner to the bottom right corner
without touching a circle.

If a circle is covering either the starting point or the end point, we
can abort immediately, so we'll assume they don't.
We will create a graph consisting one vertex for each circle, and one vertex for the south east edges of the rectangle, and one vertex for the north west edges of the rectangle, in other words:
$$V = \{ v_c \mid \text{$c$ is a circle} \} \cup \{v_{\text{se}}, v_{\text{nw}} \}$$
Then we create the edges as follows:
$$E = \{ e = v_{c_1} v_{c_2} \mid \text{$c_1$ and $c_2$ overlap}\} \cup \{ e = v_{c} v_{s} \mid \text{$c$ touches side $s$}\}$$
In other words an edge between two objects if they overlap
geometrically.
If the south east sides are connected to the north west sides, then
it means that you cannot travel from the lower left part of the
rectangle to the upper right part of the rectangle.
In other words, you only need to see whether $v_{\text{se}}$ is
connected to $v_{\text{nw}}$ in your constructed graph, which would
isolate the starting point from the end point.

Yes instance

No instance (south is connected to west)