As usual with legal issues, your understanding is completely wrong, as aquagreen explained. First, the Supreme Court had nothing to do with this case. It was an NLRB decision which was appealed to the Second Circuit Court of Appeals, and ended there. The Board (what you referred to as the "lower court") did find against the union, but the seniority provision portion of Board's decision was not overturned by the appeals court.
Your ignorance about basic civil procedure aside, your reading is also way off base concerning what the Teamsters case stands for substantively. It does not, as you imply, stand for the proposition that a union cannot resolve seniority disputes and that the employer must somehow be involved in the process. Rather, the key to understanding the case is to realize that the issue there was that the seniority list the union came up with was illegal under the NLRA in the first place. The case states that a CBA provision permitting a union to resolve senoirity disputes is not enforceable when it might result in a seniority list that violates the NLRA. Correct me if I am wrong, but I do not believe that is the issue right now with the LCC pilots -- there is no CBA provision in either the East or West contracts which says, "In the event of a seniority dispute, ALPA will determine the outcome." (Additionally, Teamsters did not have to do with a merger of two companies' lists -- another distinction from the present matter.)
Even if there is such language, don't forget the key to undertanding the Teamsters case: the list itself violated the NLRA. In Teamsters, the seniority list was illegal under the NLRA because it discriminated on the basis of union membership. In that case, some employees were not union members when they began working for the employer, but joined the union later. The union based seniority for those people on when they joined the union, rather than on the beginning of their employment,. The Board, later affirmed the Second Circuit, found that violated the NLRA because it illegally encouraged union membership in violation of the NLRA and discriminated on the basis of union membership.
Now, where the Teamsters case may be able to help East (and which USA320pilot seems to have missed) is that it does vaguely stand for the proposition that when a union is going to merge seniority lists, it should do so based on DOH. However, the case is really only on point when the union's choice is to merge lists on the basis of date of hire OR on some basis that violates the NLRA (or, by logical extension, the RLA, unless there is some RLA case out there stating something different). The case does NOT say that when the choice is date of hire OR some other method that does NOT violate the NLRA, date of hire should still be used. The NLRB and the courts got involved not because the union did not use DOH to determine seniority, but rather because it used an illegal method. In other words, Teamsters appears to leave a union free to determine other, non-DOH bases for determining seniority, as long as the NLRA (or RLA) is not violated. Therefore, if East is going to hang its hat on this case, it would have to argue that the Nicolau award violated the RLA. I don't see how they will do that.