You need to consider the whole system, and not just the radiator cap in isolation.
Once upon a time, all cars ran unpressurised cooling systems, but as the engineering improved, pressurised systems were introduced to increase the boiling point of the water/coolant. Initial efforts were usually limited to 4 psi, as with XK cooling systems, but that involved upgrading the entire cooling system to be reliable and not subject to failure at 4psi. Thus radiator cores, all hoses, hose clip design, pipe design etc had to be improved to withstand 4 psi. The pressure cap ensures the cooling system doesn’t exceed 4 psi, and thus exceed the design of the cooling system. Later on, as there were greater and greater demands on the cooling system, all aspects were improved, allowing higher operating pressures without cooling system failures, thus the 7, 13 and 15psi caps that limited the operating pressure to 7, 13 and 15psi. Its not an exact science, and levels of confidence come into it. I personally will always stay with a 4 psi cap on my XK140, albeit at a touch would probably be able to get away with 7psi, but I wont. I definitely would NOT ru with a 13 or 15psi cap, as that would be extremely high risk of cooling system failure.
Many years ago now I had an XJ40 saloon, one weekend fitted new top hose from a local Jaguar supplier, who had sourced 1/2 price top hoses of same shape/size ex Volvo. Car started up and ran OK, so drove around to a friends, parked car on his drive, went inside and came back to see a gushing pool of green coolant all over the drive way. The new Volvo hose had split along the entire top, hose-clamp to hose clamp.
Seems the Volvo hoses were speced/made for their 7 psi cooling system, and not strong enough for the XJ40s 13/15 psi designed system.
So unless you are prepared to risk all aspects of your XK cooling system, or prepared to upgrade the pressure rating of everything pressurised in system, stick with the original 4psi radiator cap, and hope that any aftermarket hoses etc you buy are well made/good for 4psi. With XKs one of the first places an over pressurised system will fail, apart from cheap hoses, is in the heater radiator core. These heaters were not designed/built for much more than 4/7 psi.
If your XK is overheating, don’t blame the radiator cap - find the real problem and fix that.