Count number of linearly independent subsets of columns of a binary matrix
Snook, Counting bases of representable matroids shows that counting the number of maximal linearly independent subsets of columns is $\mathsf{\# P}$-complete. Your problem is probably also $\mathsf{\#P}$-complete. This suggests that you need an enumerative approach.

On the other hand, you can efficiently (in theory) approximate the number of maximal linearly independent subsets arbitrarily well, see for example Anari et al., Log-Concave Polynomials II: High-Dimensional Walks and
an FPRAS for Counting Bases of a Matroid (might be an overkill).