It is my understanding that 6 bar is the design pressure (max pressure) of the main tanks. I've seen a higher design pressure of 8.5 bar reported for human rated missions. Pressure at the engines will inevitably vary.
It is known that Starship uses "balloon tanks" that require some minimum pressure inside them to prevent buckling under structural loads. We'll call this X because as far as I know Spacex haven't published anything. So pressure must be greater than X bar and less than 6 bar at all times
The height of the O2 tank on the starship is about 16m and the density of oxygen is about 1140 kg/m3. Gravitational acceleration is 9.8m/s2 so the pressure within the O2 tank varies by 16 x 1140 x 9.8 = 178,752 Pa = 1.79 bar from bottom to top due to hydrostatic pressure while Starship stands on the pad. The height of the O2 tank in the booster is about double this, so there's about 3.6 bar variation from bottom to top.
When the stack takes off, acceleration is about 0.5g, so the hydrostatic pressure increases by 50%. That is, the hydrostatic pressure at the booster engines jumps by 1.8 bar, from 3.6 bar to 5.4 bar. This does not include the ullage pressure at the top of the tank, which must be added on. Assuming the ullage pressure is 0.6 bar, we get 0.6+3.6=4.2 bar at the booster engines at ignition, and 0.6+5.4=6 bar at launch. Having done this calculation, I immediately wonder if the bottom of the booster has thicker material to handle pressures over 6 bar.
From this we can conclude that the raptors can tolerate a range of inlet pressures. The minimum pressure will be determined by the need to avoid cavitation at the pumps. Typically about 0.1 bar above the boiling point pressure of the liquid being pumped is required for rotodynamic pumps, corresponding to a liquid depth of about 1m, but it can be lower. More info can be found by googling NPSH (net positive suction head). The max pressure is known only to SpaceX but 10-20 bar is easily achievable without any real cost or weight impact.
Performance of the engines at ½ thrust has nothing to do with pump inlet pressure, and everthing to do with chamber pressure. The pumps are designed to deliver 230-300 bar to the chamber at max thrust. There are control valves to regulate the flow/pressure at turndown. Provided the pumps aren't damaged by cavitation (by operating at too low a pressure) or overpressure (extremely unlikely) they will deliver to the combustion chamber whatever pressure is required.
Somebody may know the design pressure of the header tanks (if different from the main tanks) but doubt anyone (not even Spacex) knows what pressure they will run the header tanks at for Mars approach yet. They will be figuring out the pressure profile of the header tanks for earth landing at the moment. On starship pressure is maintained by autogenous pressurization (injecting boiled propellant gas back into the tanks to fill the volume left by the used propellant) and there are a number variables to consider to find the optimum.
