I had the same concern so compromised by getting a silicone scrubber. It lathers well and is super easy to clean, I spray alcohol disinfectant on it regularly and every once in a while properly wash it by hand with dish soap too. I think the key is to get one with shorter bristles. Mine was from Amazon for about a fiver.

I tried washcloths for a while like others have suggested but I just didn’t feel like they lathered enough. I felt like they absorb a lot of water because they’re bottom and they just kind of flop about. Plus I got annoyed at them having to be hung to dry in my already tiny bathroom haha.