The pump doesn't mean anything aside form increased blood flow in that area. It doesn't mean you've stressed them enough to grow, the pump is not a good gauge.
You can drop the bies all together as long as you have 3 main movements for back.
1. Rows from the floor (each rep) with your back at 90deg to the ground (Pebdlay rows).
2. Heavy deads. (After my last set I have to sit down because I am out of breath and feel like I am going to pass out).
3. Weighted Chins 3 sets of 5-6.
All the above will give your bies all the stimulation you need to grow.
If you really want to feel the pump, as I agree it does look nice in the mirror, simply do 2-3 sets of straight barbell curls sets of 8-10. That will get the blood flowing and give you the nice pump you like, lol But it's not necessary.
If you have these 3 movements (Deads, Pendlay rows, weighted chins), you don't need to waste time doing anything else like cable pull downs, seated rows, or any cable machine, with heavy deads your traps will grow like mad, so no need for shrugs etc. Do these 3 movements in the gym, focusing on adding weight to the bar every week will save you time and get much better results.
Add in Some squats and you're set.