As you've already surmised, this is not the radiator that Jaguar would have fitted to your car on the production line (i.e. the Marston fluted side radiator) but it's worth highlighting that many 3.8 cars would have had their Marston rads replaced with the 4.2 version
by Jaguar Service Centres when owners came in with cooling/radiator issues after the 4.2 cutover. When you eventually do come to replace the radiator, it's then a matter of choice as to whether you want to replace it with a correct period-look radiator such as those supplied by Fosseway (which is the route I took with my '64 OTS) or go with a 4.2 version (which would do the job but is not as attractive from a period perspective).
In the meantime, I fear the days of your radiator are numbered. Even if the pin-hole can be temporarily fixed by Barrs, it remains an area of weakness. A further interim option to look into is the pressure rating on the filler cap. These were uprated in March 1963 (Service Bulletin D.8 / Spares Bulletin N.3) to a 9lb pressure rating to allow the water temperature to increase further (without boiling) prior to escaping from the header tank through the pressure relief hose. If yours has a 9lb cap, you could reduce it to 7lb thereby reducing stress in the tank when the water is very hot.
