Fearful, however, of losing this first and only opportunity of relieving my grief by imparting it, I, after a disturbed pause,
contrived to frame a meagre, though, as far as it went, true response.
The highlighted phrase has the very similar function of participial constructions. But fearful is not a participle, then what do you call this phrase?
It’s an adjective phrase, a phrase headed by an adjective; or an adjectival phrase, a phrase which acts as an adjective.
But that is something of an evasion. You are exactly right in seeing this as very similar to a participial clause: Fearful of losing functions exactly like fearing to lose.
What this demonstrates is that some adjectives take objects (or arguments or complements, whatever you want to call them) just like verbs, and form clauses in which the nouns they modify act as subjects. Some examples are desirous, responsible, reluctant—note that all of these derive from verbs.
And in fact there is no real difference between an adjective phrase of this sort and a participial phrase, for a participle is a verb form acting as an adjective. It’s only convention which distinguishes them.