Yes, you can switch between the two modes in code.
Or you could use :
- an LCD module on i2c, 2 wires needed for output;
- an attiny 2313 : 18 input/output pins.
Even better, why not code in assembly and use flash as a "tape" ? I.e. the normal mode for these ucons. I'm not really sure I understand what you are trying to achieve here. You still have to program the sram right ? Do you copy from flash ? What do you mean by Turing-ish ?
PS: technically, I think you only need 3 pins for 8 bits, if you do it right ...