Hi, Gian and thanks for the assist Jay J. Gian, the only thing that I question is using the steel wool. The little fibers can get down in the wood, especially on oak. It can later rust and cause dark spots, but I don't think that this is what's causing the problem now.
Perhaps the problem is still dust in the surface? After you sand with the 220, tack rag again.
I have never run into a problem with lumps and bumps in poly, but I do use a stir stick. Most brushes leave so many bubbles that I don't worry about air bubbles from stirring or mixing. I do go back over poly a couple of times with almost a dry brush to smooth the surface and get any bubbles out. Do this very lightly--3 hairs and some air is all you need on the surface. Usually I do have a few bubbles or dust spots after the first coat so I sand and tack cloth before the second coat. You should do this in any case to give the finish some "tooth" for the first coat to stick to the second coat. Depending on the viscosity of the poly, I will either 220, or sometimes 400 dry.
If you feel that the bumps might be solids from the poly, you can get a paper strainer at the paint store but, honestly, I have never run into this. By the way, it is water that raises the grain so water based poly is succeptable to this but the sanding between coats solves any problem.
I have run into one thing that also might be the culprit. I take an old coffee can and pour some thinner in it then cut an "X" in the plastic lid, stick the handle thru the "X" and let that suspend the brush in the thinner between coats. I don't know what causes it or the chemistry, but I have gotten yellow blobs (looks like some kind of wax, sort of) that "grow" around the outside and bottom of the can. These are very evident and I just wash the brush well and start on a new coffee can. Perhaps this is what's causing your bumps. Rinse the brush in fresh thinner before you apply the second coat. This usually takes a couple of days to a week to start but this is my guess for the cause of the problem, particularly if you are doing one door right after another.
Hope some of this hits the right note for you. Henry in MI