cf. For regular matroids: deciding whether a regular matroid has a circuit of length at most k containing two fixed elements is fixed-parameter tractable. arxiv.org/abs/1108.5457

If true, then we may assume that $V_1=V_2=V$ without loss of generality.

There are related or identical questions on math overflow. mathoverflow.net/questions/67960/… mathoverflow.net/questions/16393/…

Perhaps, you want to check out the concept of linking systems due to A. Scrijver (1974). It is equivalent to matroids.

I am pretty sure that the well-quasi-ordering result of binary matroids would imply the well-quasi-ordering of graphs under odd minors. I also remember hearing Tony Huynh talking on structure of "group-labelled" graphs with a forbidden minors. Perhaps, though not everything is published, it must be known.