CTEK only recommend using 1 x 2.5m extension with their maintainers. That said, I have 2x 2.5m extensions to get from the garage to the car and have no problems with charging.
Changing the cluster is not a particularly difficult job to do, a few screws and a couple of connectors. I think though, the cluster needs to be programmed to the car.
May be best to talk to a Jaguar Independent Specialist first before going ahead.
12.4v to start with is a little low, fully charged is 12.8v. With the engine running you should see 14.3v to start with and then reducing to 13.8v.
If your car is diesel then the charging from the alternator may be a little different to that of the petrol version.
When you lock the car place the fob at least 2 metres away from the car, does the car then stay locked when you try to open it. Not trying to teach you to suck eggs but some don’t understand how the system works.
First thing Mark, keep it simple. Check the battery voltage and fully charge if need be. 12.8v is fully charged. Check the alternator output. Check your ground points in the engine room are all clean and tight. Reset the codes and then see what comes back.
First I’ve heard of this happening on a F type. The alarm chirp sounder may need replacing, the rechargeable batteries can cause problems. Most likely fitted under the drivers side front wheel arch though I’m not 100% sure of that.