I have done a bunch of reading on the problems with the HUEI system on the Cat C-7. In all the cases the owner was running regular oil not synthetic. It was also pointed out that the HUEI system needs to be kept as clean as possible inside. I think I would run the new High efiencey filters such as the Cat 1R-1807 and 1R-1808 filters contain an improved, higher efficiency filter media. The Cat 1R-1807 and 1R-1808 filters offer better filtration efficiency than standard oil filters, improved engine cleanliness and a longer component life. This media is designed to increase contaminant capturing efficiency while maintaining excellent dirt holding capacity characteristics.
The other thing I noted was the bearings in the pump had failed from inadequate liberation. This tells me I would run a synthetic oil to better lubricate the pump bearings. I would run Mobil 1 or Rotella T-6 5-40 oil.
You need to have oil pressure as soon as possible to the bearings in the pump.
Bill