No, the term is unlikely to be related to Kant, and self-reference did not even enter the picture at the time Russell introduced it. It was brought up by Poincare, who appropriated the term later and shifted it towards its modern meaning.
Variations on the word "predication" in logic go back to Latin translations of Aristotle and elaborations on them by medieval logicians and grammarians. It referred to attaching adjectives and the like that could be said of the subject to it in a sentence. By the end of 19th century it came to mean, as now, linguistic representation of properties (or relations), but the modern distinction between extensional sets and intensional classes specified by them was still shaky, if made at all.
In his 1906 paper On Some Difficulties in the Theory of Transfinite Numbers and Order Types Russell tried to figure out why some set/class theoretic definitions lead to paradoxes and to find conditions on propositional functions that would preclude them. He proposed three tentative solutions, which are termed zigzag theory, limitation of size and no classes theory, and introduced the term "predicative" as follows:
"In this paper I shall use the words norm, property, and propositional
function as synonyms... We have thus reached the conclusion that some norms (if not all) are not entities which can be considered independently of their arguments,
and that some norms (if not all) do not define classes. Norms (containing
one variable) which do not define classes I propose to call non-predicative;
those which do define classes I shall call predicative. Similarly, by extension, a norm containing two variables will be called predicative if it
defines a relation; in the contrary case it will be called non-predicative.
Thus we need rules for deciding what norms are predicative and what are
not, unless we adopt the view (which, as we shall see, has much to recommend it) that no norms are predicative.
"In the zigzag theory, we start from the suggestion that propositional functions determine classes when they are fairly simple, and only fail to do so when they are complicated and recondite. If this is the case, it cannot be bigness that makes
a class go wrong; for such propositional functions as "x is not a man "
have an exemplary simplicity, and are yet satisfied by all but a finite
number of entities. In this theory, as well as in the theory of limitation
of size, we define a predicative propositional function as one which determines a class (or a relation, if it contains two variables); thus in the
zigzag theory the negation of a predicative function is always predicative." [emphasis mine].
Kant is not mentioned in the entire paper, and the definition does not refer to "existence". The discussion is even further removed from Kant's objection to predicating existence of a subject as a "real" property. That had more to do with existence being a second order property (in modern terms) applicable to subjects with all their (first order) properties already attached ("we do not add a thing to the concept of a thing by further stating that the thing is"). In essence, Kant's idea was adopted by Frege and Russell in their predicate calculus, see What are the counterexamples to Kant's argument that existence is not a predicate?, but that had little to do with the paradoxes.
From context, Russell is instead concerned with which propositional functions stand for "entities" that can be admitted into logical calculus fit to build mathematics. So "predicative" seems to be intended as something like fit for predication in its traditional Aristotelian/scholastic sense. This is supported by Russell's remark that "a statement about x cannot in general be analyzed into two parts, x and what is said about x".
According to Feferman's historical survey Predicativity, Russell was not very successful with his proposals, and further elaborations on predicativity appeared after Poincare suggested the vicious circle principle (self-reference) as the culprit behind the paradoxes the same year in Les mathématiques et la logique. Russell accepted the suggestion, more or less, and only then the term began to gain its more specific modern content. Poincare also preferred "impredicative" to "non-predicative".
