Some more info to share on Moricz and Zali (I don't know what you have already):
- Zali died 1894.03.31, which makes sense given that Moricz remarried that same year to Zali Dicker
- they had at least 7 children: Adolf (1876), Jetti (1878), Gergely (1883), Herman (1886), David (1888-d.1890), Bernat (1890), Jakob (1894).

It's possible that the Ignatz you're looking for was another son of Moricz and Sally Kline, who married Rose Gross (daught. of Sigmond Gross and Etta Altman) in New York 1904 (I see conflicting dates for the marriage, either April 15 or May 29). I don't know what his birth date was, but his marriage record clearly indicates Mor and Zali as parents.

I found a draft card from 1942 for an Ignatz Reichard born in Hungary in 1878, who was unemployed and living in the Bronx. He died March 8, 1946 in the Bronx. But I can't say if this is the same person as the Ignatz who married Rose Gross (the contact person on this second Ignatz's draft card was not a wife).

I've got several other Ignacz Reichards in my Sataroljaujhely lists, but I'm guessing these are too old to have been the person your cousin remembers from the 1930s (these others were born between 1858-63).

I haven't been in touch with any Reichards from Brooklyn.