If the BSD rank conjecture were false, then the simplest counterexample might be an elliptic curve with algebraic rank 4 and analytic rank 2. This could be established for a specific curve by rigorously numerically computing the second derivative of the L-series at 1 to some number of digits and getting something nonzero (which is possible because elliptic curves are modular - see work of Dikchitser). This is a straightforward thing to do computations about and there are large tables of rank 4 curves. This is also exactly the problem I suggested to the OP in grad school. :-)

In number theory doing these sorts of “obvious computational investigations” is well worth doing and led to many of the papers I have written. I remember doing one in grad school and being shocked when we found a really interesting example in minutes, which led to a paper.