The paper Enumerating Matroids of Fixed Rank will probably be of interest to you. Although it does not address the question of rapid mixing, I suspect that the methods they use to obtain their bounds on the number of matroids of fixed rank $r$ on an $n$ element set can be transformed into fast sampling algorithms. Their work relies on the classic result of Knuth that you already mentioned, together with some new ideas. In particular, they prove that every matroid $M$ can be reconstructed from $n$, $r$, and a certain antichain $\mathcal{V}(M)$.

The paper Enumerating Matroids of Fixed Rank by Pendavingh and van der Pol will probably be of interest to you. Although it does not address the question of rapid mixing, I suspect that the methods they use to obtain their bounds on the number of matroids of fixed rank $r$ on an $n$ element set can be transformed into fast sampling algorithms. Their work relies on the classic result of Knuth that you already mentioned, together with some new ideas. In particular, they prove that every matroid $M$ can be reconstructed from $n$, $r$, and a certain antichain $\mathcal{V}(M)$.