T&T: Perkins 6.3544 NA Problem(s)

[Bob England]

Increasing oil would point to the lift pump, usually, especially if that were
the only symptom. The blue smoke kinda muddies the diagnosis tho. It sounds
more like a bad injector. leak down could be causing the oil increase. The
input shaft seal in the injector pump going bad will put fuel in the oil, but
not generally cause the over rich condition you are describing. Diesel in the
oil could be allowing it to get past the rings into the combustion chamber,
but that would indicate a fairly well worn engine, at least loose oil rings.
The blue smoke does point to burning engine oil, which could be the result of
fuel in the oil. I'd try replacing the lift pump again, or just try bypassing
it with an electric unit temporarily and see if the problem goes away.