Suppose I have a pair $(X,\mathcal{F})$ such that $\mathcal{F}$ is an intersection closed family of subsets of $X$ for which $X\in \mathcal{F}$ is there a technique I can use to determine easily if $\mathcal{F}$ is the set of flats of some binary matroid grounded on $X$?
Preferably one that doesn't involve me reconstructing a closure map, using it to find independent sets and then verifying that this is a matroid and that its circuits all satisfy a number of properties i.e. just a test on sets in $\mathcal{F}$ would be good.
First test if $\mathcal{F}$ is the set of flats of a matroid by verifying the three flat axioms (see, e.g., the matroid wikipedia page's section on flats). By what you've given about your set of closed sets you only need to check the third condition there.
If your closed sets are the flats of some matroid then construct the associated lattice of flats and confirm that every interval of height two has at most five elements (see, e.g., the alternative characterizations section of the binary matroids wikipedia page).