What Do We Call: Having Been + Present Participle?

participles

I am inquiring in terms of classic prescriptive grammar. As a preface, please consider the following usage of the perfect participle.

I, [having played] well, am proud of my game.

The participial phrase, having played well, is an adjectival modifying the subject, I.

My question concerns constructions of "having been" + the present participle, like:

I, [having been playing] well, expect to win.

To my eye, having been playing is a participle, and the participial phrase, having been playing well is an adjectival modifying the subject, I.

Is having been playing a form of participle? If so, what do we call that form?

The needed nomenclature can be chosen as that provided in §§ 3.54 and 3.56 from "A Comprehensive grammar of the English language"; and as well a system for naming all complex verb forms is given.

3.54 There are four basic types of construction in a complex verb phrase:
Type A (MODAL) consists of a modal auxiliary + the base of a verb: eg: must examine.
Type B (PERFECTIVE) consists of the auxiliary HAVE + the -ed participle of a verb : eg: has examined. (Traditionally the term PERFECT has been frequently used instead of PERFECTIVE.)
Type C (PROGRESSIVE) consists of the auxiliary BE + the -ing participle of a verb : eg : is examining.
Type D (PASSIVE) consists of the auxiliary BE + the -ed participle of a verb: eg : is examined.
These four basic constructions also enter into combination with each other:
     AB :      may have examined
     AC :      may be examining
     AD :      may be examined
     BC :      has been examining
     BD :      has been examined
     CD :      is being examined
     ABC :    may have been examining
     ABD :   may have been examined
     ACD :   may be being examined
     BCD :   has been being examined
     ABCD: may have been being examined