I had 3406B,C,E also had C15s, and one C12. Along with Cummins and Macks.
Here is my advise on your new reman.
Use CAT filters right from the start. Use whatever oil the reman company specs.
Do an oil analysis at 5K or sooner.
Look for fuel, coolant and silicon. On a reman engine what can get you in trouble is improper installed injectors, pump, leaking head gaskets, liners, waterpumps, oil cooler and air piping.
Once you have figured out you have a solid engine then you can move around with oils and filters.. but all that kitty needs is fresh fuel and clean air.
I have used NAPA, Baldwin, Luberfiner, CAT and Mack Filters..
Rotella, Spirax, Valvoline, NAPA, Traveller and Mobil oils...
None made any real difference to the bottom line.