I think you can achieve that in the SQRL app (I don't know if it does in practice). The app would have multiple private keys with one flagged as active. You would toggle which is active before reading the QR code (or tapping on it if on the same device).

I believe technically it "computes" the private key on the fly as you need it (it's derived from the master key plus other factors to make it unique per site/user).

A user inputted string can be provided at login. Together, the master-key, the inputted string, and the site domain are combined to regenerate the site-specific public-key private-key for that sub-identity.