Nox had the same 32-bit problem. You need (the last 64-bit beta for Catalina).
BlueStacks recommends at least 8GB of RAM for a smooth experience. If your Mac only has 8GB total, try to limit the emulator to 2GB or 4GB in its settings. bluestacks mac catalina
For the best performance on Catalina, look for BlueStacks Version 4.240 or higher (specifically optimized for macOS 10.15). Nox had the same 32-bit problem
System Extension Suddenly "Blocked" on my Mac : r/BlueStacks bluestacks mac catalina
A common quirk on Catalina is that the "Allow" button in System Preferences > Security & Privacy might not appear immediately. You often have to launch BlueStacks multiple times or wait for it to "trigger" the macOS security request.