You have to find a way to reduce the size of your simplicial complex. Some algorithms based e.g. on discrete Morse theory can do that fairly rapidly, but they don't have guarantees on the amount of size reduction. I don't think there exists faster algorithms for approximate Betti numbers in general, but I believe it can be done if your simplicial complex is "low-dimensional", meaning that for any point, the size of the sub complex formed by vertices at distance less than $r$ grows slowly with $r$.

You have to find a way to reduce the size of your simplicial complex. Some algorithms based e.g. on discrete Morse theory can do that fairly rapidly, but they don't have guarantees on the amount of size reduction. I don't think there exists faster algorithms for approximate Betti numbers in general, but I believe it can be done if your simplicial complex is "low-dimensional", meaning that for any point, the size of the sub complex formed by vertices at distance less than $r$ from that point grows slowly with $r$.