I'll admit I don't have a CS4, but given that the pump is rated to 400 bar, I would not have any issues in feeding one a few psi of air on the input side for filling tanks to ~300 bar. Here is why . . .
This compressor is a 2 stage unit, and a rating of 400 bar implies that it has a compression ratio of ~20:1 on each cylinder, which I will admit is high. But all these air compressors use some form of check valve on each cylinder - meaning the air flows through the valve when pressure is below the upstream pressure past the valve, but can't flow back. That makes them somewhat "self balancing" in terms of load - neither stage will operate anywhere near "full pressure" when the peak output is well below the rated output.
Now if we shift our thinking from "gauge pressure" to "absolute pressure", we realize that when we turn it on at sea level we are feeding it 1 bar of air pressure (or ~14.5 psi), and to get to max pressure it is compressing that 1 bar of air ~20:1 in each stage. If we feed it air under pressure, for example 2 bar absolute (or 1 bar / 14.5 psi on a gauge), that absolutely would overload it if we ran it to max because we actually would be running it twice as high as designed - of course the motor would possibly stall, or more likely the burst disc would pop. But if we are not running it that high then the compressor really does not notice the difference in most regards because the pressure and load self balances - the power needed to pump to a given level below rated capacity would be the same. Of course this has limits - there clearly would be a level of boosting that would "lock up" the pump as there would not be enough volume for compressed air after the first stage check valve to accommodate the amount of compression occurring - but small amounts of boost would not be a problem.
Of course the big issue is heat - the compressor would be doing twice as much work on each stroke, thus generating twice the heat. That in itself is bad, but if we operate well below rated capacity then the heat load is still far below the max design constraint anyways - we probably would not generate more heat pumping 2 bar absolute (one bar of boosted) air to 3000 psi than we would pumping one bar gauge (base operation) to 5800 psi.
So I think it is all about striking the right balance with it. Personally, If I had the set up that
@BlackICE has I'd give a a try of boosting the input air maybe 4 psi or so and see how it performed - but only for filling no higher than 310 bar/4500 psi. If it worked well and I was filling to only ~200 bar I'd consider maybe up to about 7 psi (1/2 bar). Doing so would cut the fill times by ~25% and ~50% respectively.
I can say that while I don't have one of these pumps, I did once accidentally make a big mistake with my Shoebox compressor (those use a shop compressor as the first stage, and then two more stages internally to get to 4500 psi). I was testing theories out and had disabled the original mechanical pressure switch to see how that impacted controlled fill times (since the way it worked resulted in a loss of compression stroke distance) and I did not have the fill valve on the 45 min tank opened properly. The compressor ran fine with no signs of stress, and the only way I noticed the error was when the burst disc blew at about 7500 psi (517 bar). Scared the crap out of me, but all was fine with the compressor - swapped out the disc and the unit was no worse for it. That same compressor now has over 300 hours on it and still runs like new . . . .
