Here you've hit on a subject of particular interest to me. I would like to carry a fountain pen on my person, and I do sometimes have to fill out a form that was printed on cheap copy paper. Sometimes I have to write on both sides, even. I've been actively researching inks that could perform well in such situations.
This was also one of my intentions after starting to use fountain pens. The paper I was targeting was not copy paper per se. Instead I was looking for inks which were writing on notebooks like Moleskine and note blocks like Mead Cambridge series.
I've started my search with Lamy Blue and found that it behaved reasonably well but, using an ink which can easily erased and forged was a big red flag for me. Lamy's black was also very well behaving and was pH neutral (it's almost like water in that regard). On the positive side, it's relatively water resistant. Then I had to pause my research because Moleskine started to use much thinner and cheaper paper. On top of it, the new note block I bought for the office didn't like the ink that I'm using (it was actually repelling it).
For a long time Noodler's Black was my standby for this, and I really don't consider it a dangerous ink at all. There's some question about Noodler's in general possibly being hard on rubber sacs, but I don't need to carry a pen with a rubber sac. Noodler's Black is not pigment-based. It is highly saturated, but really…
Noodler's black is considered as a soft and safe ink in that regard. It's marketed as pH neutral so, it should not melt pens like first batches of Baystate Blue. On the other hand, I've no rubber sac pens and, I cannot comment on that front. I've recently purchased and started to use it. It's a well behaving ink with very controlled flow on my Lamy Al-Star. It started grey-ish but, its saturation is increasing as the feed saturates. It's a cellulose reactive formula, so it should wash out of everything except cellulose with relative ease.
I've found out that Kaweco's Brown has this strange property. While it's not waterproof by any means, it doesn't stain anything except paper. Also it's a well behaving ink. It doesn't attack the paper, but calmly dries on top of it.
Despite my defense of Noodler's, I have moved away from that brand lately. The last few bottles that I opened have become more and more inconsistent, so I began looking for alternatives.
Actually, I have a soft spot for Noodler's for their colors, advanced chemistry and its strange sincerity. What kind of inconsistency has surfaced in your inks? Maybe this should be communicated back to Nathan or Noodler's directly?
BTW, If I remember correctly, Noodler's "forgery resistant" inks has batch to batch variations intentionally. Is this property applicable to your bottles?
I also began testing various inks for performance on cheap, absorbent paper. This led to a few surprises.
Many of the old standby inks that you've mentioned turned out to be "not bad" in this respect. Pilot Blue and Aurora Blue and Waterman Mysterious Blue worked a bit better than your average ink. Pelikan 4001 Brilliant Black was pretty good. Iron Gall based inks, like Salix and Montblanc Midnight Blue, were even better, but those IG-based permanent inks are also higher-maintenance.
I'm glad that these inks performed really well. Maybe you should give Lamy Black a try? As far as I understood you prefer black inks as daily drivers. I'm a blue & blue/black person so, I cannot comment on many black inks. Sorry about that.
Then I discovered Diamine Eclipse, an almost-black ink that is non-permanent but outperformed even iron gall inks at not soaking through the page. I've also been pointed toward Diamine in general by at least one pen repairman as a "safe" ink brand that should be compatible with latex rubber. Well, Diamine have been in business for 150 years. They should know a few things.
Diamine also has a nice spectrum of colors and, as you've said, they're one of the oldest players in the ink arena. I have a couple of bottles (Oxblood, Denim blue), but didn't try them yet. I also didn't read anything bad about their inks but, I just don't have the experience to back these claims.