 Solution

Every finite partial isomorphism between models of $T_1$ extends by one point. If the new point belongs to a class already represented in the domain, choose an unused point in the corresponding target class. Otherwise choose an unused point in the other target class. Both classes are infinite, so the choice is always possible. The back-and-forth method shows that tuples with the same quantifier-free type have the same complete type. Therefore $T_1$ has quantifier elimination.

Solved by gpt-5.6-sol high.