Looks good!
The "compression ratio" issue is not a problem - you won't be overloading the second stage, as these pumps basically balance the load through reed type valves. Compression load is basically split between the two cylinders dynamically as pressure increases - these are not fixed closed cylinders like in a car engine - the pressure "leaks out" into the next stage when the pressure rises above what the valve has holding it closed. So you are actually reducing the relative compression load on the cylinder by feeding under pressure.
Think of it this way - when boosting from 1 bar to 300 bar, that is a 300 times increase in pressure in two steps. The square root of 300 is bit over 17, so at 300 bar both cylinders are basically compressing at that level. By feeding it 6 psi, you are feeding it about 40% more air, so it is starting at 1.4 bar. We have to "normalize" that starting pressure out in a two stage system, so booting that 1.4 to 300 bar would be a 214 times increase in pressure (300/1.4). So now the square root is under 15, so less actual compression takes place (40% less, although the pressure is the same at the output.)
This is a good thing, but I would not go overboard on it - I would not go over 2 bar absolute, or about 15 psi gauge, on the input side just out of caution. That said, you could test the stress on the compressor itself indirectly by measuring the temperature of the heads over time at different input pressures. There will be a balance point where the load is going down initially, but then rises as the total amount of air being compressed on each stroke leads to higher heat load because the total amount of air being compressed on each stroke is higher too. But you should be fine where you are now.
