It defo sounds like a fuelling issue, all the parts need to be std to work with a std chip, why has it got a FSE reg on, these are crap, and the rate of increase could be substantially different from a std cossie one.
The voltage check you did was a good idea but unfortunately this MUST be done whilst under maximum load, the 12v or whatever figure you had can easily half in value once a load is placed on it.
Personally I would just rewire using the original live feed at the pump top trigger the new relay, I would take a earth direct from the battery negative, as well as the live from there, through a fuse by the battery.
You can use 28/030 wire, as this carries 17.5 amps continuous so is plenty man enough for the job.
As a short term evaluation you could just wire direct to the battery supply and try that to see if it is any different.
tabetha