Sam,
With all air pistols, the air tank is limited in volume, not just the GK1. Some are air pressure regulated. These shoot consistently, but this comes at a cost of power and available tank volume simply because the regulator takes up available tank volume, which is already limited in a pistol. The unregulated pistols, like the GK1 are more powerful. So, to manage this POI issue, it is necessary to recharge the air tank after every magazine. As fishing43 said, fill slowly. I actually fill twice to counteract the tank pressure drop from heating and cooling. So, after shooting one magazine, I charge the pistol to 300 bar. I then reload the magazine. By this time the pistol tank has dropped from 300 bar to about 270 bar as the tank cooled while I was reloading the magazine. I then recharged the tank back to 300 bar from the loss of tank pressure due to tank cooling. My GK1s are .25 and have 17 pellet magazines.I set my power to slightly more than 700fps because it provides the best speed consistency over the 17 shot magazine. After one magazine has been shot the tank pressure is about 210 bar.
The next issue you have to contend with is the usable volume of your storage air tank, because only a small percentage of that tank volume is usable before that tank pressure needs to be recharged.. You stated that you use a 2 liter bottle for that purpose. That is way too small for that purpose. I use a 12 liter bottle and my 12 litter bottle gives me about 7 or 8 GK1 charges. I am considering getting a 40 liter steel bottle for that purpose. Of course, it would not be very portable, but it would save the number of recharge sessions required over say a year, as I shoot mostly at home and steel bottles are cheap. I have 3 12 liter bottles. Two are aluminum SCUBA and rated to 220 bar. I also have a 12 liter ACECARE CF bottle, which I use with the GK1s and it is light and handles my away shooting requirements. As a final note on this subject, all these tanks are expensive and the larger tanks require a higher volume compressor with a higher duty cycle rating that is also expensive. The bottom line here is that high pressure air (HPA) is the most expensive part of PCP ownership. Way more than the guns themselves.