Thonientaka:ion's given names do seem to have varied a lot. I remember him using at least Henry ( or Hendrick ), Philip, and Hanyost, but the nickname seems to have been more consistent. You're right that Trathoop is probably the same name as Blatcup, and I hadn't noticed this ( I've been writing it as "Trathroop" for some reason ). Oneida doesn't have B, and so conceivably Blatcup might have been pronounced Tlatcup ( or Tratcup, since Oneida docs and hymns from the 1800s still have R instead of L, like Mohawk ). That would then be the source of Trathoop, which is otherwise not a first or last name in English, German, Oneida, or Mohawk. Possibly it is written "Tratkoop" in the original list of losses sustained and the K looks like an H.
I would guess that the name Blatcup wasn't used by Moses because it was the nickname of the father rather than the surname. I would still probably take Blatcup to be Moses Schuyler's father ( rather than the German Hanyost Schuyler ) because it's mentioned in the Draper manuscript a number of times and by at least two different informants, cf.
"Blatcop's (Moses Schuyler's father) Indian name was Tha-ny-en-ta-ga-yon or Old Legs: died ab't 1819, aged about 80 – a very able warrior – no chief, but a counsellor of the nation."
[ Here Jacob Cornelius says that Blatcup Thonientaka:ion was Moses Schuyler's father ]
"The mother of Hon Yerry Doxtator had a son by a German named Schuyler, who was the grandfather of Moses Schuyler. The latter's father was nick-named Platcoff, meaning Flat Head, a small man, who died about 1822, very aged (Han Yost Schuyler)."
[ Aunt Polly Maytop says that Moses Schuyler's father's nickname was Platcoff ( Standard German Plattkopf in a few other sources from the Revolution ), and if the grandfather's last name is Schuyler then the implication is that this was Blatcup's last name also. Possibly the name in brackets signifies that "Han Yost Schuyler" was Blatcup's European name, but it's difficult to interpret. If so then he was born Thonientaka:ion, used the European name Hanyost ( or Philip, or Henry ) Schuyler, and was nicknamed "Plattkopf", which seems to be what people actually called him ( i.e. rather than Philip, Henry, Hanyost etc ). ]
"Blatcup (Moses Schuyler's father) was a Pagan Chief, & died several years before the Oneidas removed to Wisconsin – fully 80 years of age. Don't remember his war services, but he was active in the Revolutionary War."
[ Still Aunt Polly reporting – Blatcup is again mentioned as having been the father of Moses ]
"Aunt Polly knew Hon Yost Schuyler, a Mohawk German – very aged – lived at Connestota, & died about 1810 – talked Oneida fluently. Maybe he was Blatcup's father."
[ So Aunt Polly knew the German Han Yost, who lived with the Oneidas, but is uncertain as to whether he could have been the father of the other Schuyler (Blatcup) – she recognizes them as having been two different people ].
Age-wise, Chief Moses Schuyler ( Shona:kares, "Long Horns" ) is listed as aged 75 on the 1851 census, so if not an exaggeration then that would put his birth around 1776. Blatcup Schuyler ( Thonientaka:ion, "Old Legs" ) is supposed to have died aged 80 in 1821 or 1822, which would put his birth around 1741-1742, which may be exaggerated – although Hanyere's children were baptized in the mid-1750s, so if we assume Blatcup was the younger of two half-siblings then maybe it's not too far off.
The German Hanyost who lived at Oneida Castle wasn't born until 1744 though, so a problem would arise in his being the father of Blatcup ( who was born say 1742 ), or the grandfather of Moses Schuyler ( b. say 1776 ). If Jacob Cornelius and Aunt Polly are both wrong about Blatcup being Moses Schuyler's father then it's possible that German Hanyost ( b. 1744 ) was the father of Moses ( b. say 1776 ). Blatcup is the only one Aunt Polly and Jacob Cornelius associated with Moses Schuyler, though, and the possibility that German Hanyost was Blatcup's father is expressed by Aunt Polly only as a "maybe" towards the end; possibly Draper asked her and she was uncertain of it because the one individual seemed to be not much younger or older than the other.
As to who was the father of Thonientaka:ion / Blatcup, I'm not sure. If it really was a German named Schuyler then there are many candidates, since the Schuylers had been around since New Netherland and were in contact with the Five Nations early on. If Blatcup's European name was from a patron, however, then it might have been borrowed from General Philip Schuyler ( assuming Blatcup did actually go by the name Philip at some point, which the entry for the Haldimand letter of 1780 implies only in linking the two names Thonientaka:ion and Philip Schuyler; I haven't seen the original ), or maybe it was even borrowed from the German Hanyost, a European of comparable age living among the Oneidas. It would be interesting to see if the Oneida Schuyler Y-DNA is a match to the New Netherland Schuyler Y-DNA but I can't find entries for either in databases. Hope this helps anyway, I'll let you know if I come across anything more.