Guitar picks work great. They come in several thicknesses, so buy a selection. Like you say the main o-ring is a tight fit. It takes a little practice removing & re-installing. Place it flat into the groove with a little finger pressure. If you work it parallel to the groove it will elongate slightly & won't fit properly (appears too long).
On a trip I only remove the o-ring if its really necessary to properly clean it. Most of the time I can remove minor debris with it in place. Try to keep the area between the o-ring & groove clean. Something like the corner of a business card works for this. You can flick out small particles with it. I usually can do a long dive trip without removing the o-ring but do remove it afterwards for a good cleaning & complete re-greasing
I suggest buying a complete re-build kit for the housing. It includes ALL the o-rings & new stainless c-clips that go on the button shafts. Also 2 strap wrenches (plastic handle with about 20-25mm wide rubber strap). Sometimes its really hard to unscrew the port. Nexus is a great housing. Not many of us use one.