Personally if I was only getting 250m range from a gw (yes it has happened!) I would investigate why and what affecting - equipment or environmental issues - before looking to invest in better antenna.
Can you take the GW elsewhere to run checks to eliminated built environment issues?..Check existing ant & kit correctly set up with tight connections using correct connectors (e.g. sma m/f, sma-rp m/f etc.). Check test node with other TTN gw’s etc.
250m are so few that likely the antenna is not the only issue. One crucial thing is where the gateway is placed: if you put it in the middle of the apartment, table level, even with the new antenna you will not reach kilometers. Plus, checking connections is also a must.
I didn’t talk about exemptions. I referred to his bad reception capability which will not improve that much with an +6dBi antenna. “Big gain”-antennas for LoRa (with it’s huge link budget) are overestimated.
Legal depends on the output of the antenna, not the output of the transmitter. So if you suffer a huge loss in the cables and connectors a 6dBi antenna might well be legal without changing the transmitter output.
Two days before I was able to establish a LoRaWAN link of 8.7km LOS in urban area with a tiny antenna similar to THIS on a ~ +16dBm transmitting INDOOR node. The signal was received with an astonishing -137dBm. The relevant receiving gateway uses an λ/4-Ground Plane outdoor antenna. Most important for an antenna is it’s exact frequency band tuning.