Is 'ponder' acceptable as a quotative verb, for instance with inner dialogue?

I see in this matter nothing else but a question of semantics. In the OALD this particular type of construction is regularly labelled as "[V-speech]" (in the online edition "+speech"), meaning "verb followed by speech"; this classification departs from the traditional transitive/intransitive scheme as those uses ([V-speech]) are found to be neither. A different semantical context must be supported grammatically by a different grammatical category.

The examples that can be found in this dictionary
(those I could remember or glean in the various remarks on the present ELU page, which might be sufficient for the purpose of drawing conclusions)

A - First kind

say [V-speech] ('Hello!' She said) ♦ tell [VN-speech] ('I'm ready to go now,' he told her.) ♦ go (infml) [V-speech] (I asked 'How much?' and he goes 'Fifty' and I go 'Fifty? You must be joking!') ♦ continue [V-speech] ('In fact,' he continued, 'I'd like to congratulate you.') ♦ add [V-speech] ('And don't be late,' he added.) ♦ bark ♦ yell [V-speech] ('Be careful!' he yelled.) ♦ repeat [V-speech] ('Are you sure?, she repeated.) ♦ interject [V-speech] ('You're wrong,' interjected Suzan.) ♦ smile [V-speech] ♦ laugh [V-speech] ('You're crazy! she laughed.) ♦ beam [V-speech] ♦ grumble ♦ exclaim ♦ pant ♦ retort, ♦ breathe ♦ think [V-speech] ('I must be crazy,' she thought.)♦ muse (syn. ponder) [V-speech] …

B - Second kind

ask [V-speech] ('Where are you going ?' she asked.) ♦ complain [V-speech] ('It is not fair,' she complained.) ♦ exhort [V-speech][VN-speech] ♦ advise [V-speech][VN-speech] ♦ warn [V-speech][VN-speech] ♦ promise [V-speech][VN-speech] ♦ order [V-speech][VN-speech] ♦ answer [V-speech][VN-speech] ♦ request [V-speech] ♦ wonder [V-speech] ('What should I do now?' she wondered.) ♦ ponder [V-speech] (‘I wonder why,’ she pondered aloud.) …

It is clear, from the end of this list that the usage has been recognized; however, this is not the answer in my opinion.

It is clear that for the verbs of the first kind, the action is that of the reporting of an occurrence of an enunciation or, in addition to that, the manner in which this enunciation is carried out.
In the second kind there is not any more in the meaning of the verbs a concern with reporting an enunciation; this is taken for granted: in all cases the verb can be replaced by any of the verbs of the first kind (bar for an adjustment: told them/him,…)) and the sentence remains meaningful. It is not so if we look at this the other way around.
All that can be made out in the meaning of the verbs of the latter kind is a concern with the sort of quoted speech; they characterize semantically the type of enunciation that this quoted speech constitutes.

From Wikipedia, speech act

In the philosophy of language and linguistics, speech act is something expressed by an individual that not only presents information, but performs an action as well.
Speech acts serve their function once they are said or communicated. These are commonly taken to include acts such as apologizing, promising, ordering, answering, requesting, complaining, warning, inviting, refusing, and congratulating.

Note that all the verbs in the second kind are in one of those categories and that in the above dictionary only for four of the categories the corresponding verb is not "V-speech" or "VN-speech (apologizing, inviting, refusing, and congratulating). One may wonder why, as for instance the following possibilities appear justified.

• 'Sorry for that,' apologized Irene. (There is in fact an example in this book: House of Gold, Bud Macfarlane 2014.)

• 'Get in,' invited Irene. (likely enough possibility)

• 'A blunt "No",' refused Irene. (less likely than the preceding but not unconceivable)

• 'Well done,' Irene congratulated him. (One instance is found in this book: Sword of the Gods: The Chosen One: (Book 1 of the Sword of ... Anna Erishkigal 2012.)

All verbs of the second kind are of the nature of verbs that can be used for a speech act. They are all performative verbs except "wonder". ("Ponder" is not a performative either.)

From Wikipedia, Performative verbs

Performative verbs are verbs [the action of which is] carried out simply by means of uttering them aloud. When a judge sentences someone to jail time, for example, the action is completed when he or she says, "I hereby sentence you to five years in prison," or the like.

One can identify a performative verb by using the "Hereby" Test. In English, only performative verbs may be preceded by "hereby" while other verbs in the same context are unacceptable.

• I hereby wonder what you should do.
• I hereby ponder what pleasures I have found.
• I hereby think that I must be crazy.
• I hereby muse that.

From ThoughtCo, Performative verbs

In English grammar and speech-act theory, a performative verb is a verb that explicitly conveys the type of speech act being performed. A speech act is an expression of intent—therefore, a performative verb, also called a speech-act verb or performative utterance, is an action that conveys intent. A speech act can be in the form of a promise, invitation, apology, prediction, vow, request, warning, insistence, forbiddance, and more. Verbs accomplishing any of these are performative verbs.

The nature of the quoted speech construction for verbs that do not correspond to a simple reporting of a speech occurrence, because they have a meaning totally foreign to that effect, must be taken as formulating an idea related to the meaning of the quoted utterance. Otherwise, what is their role in the quoted speech construction? There is no connection. The pattern imposes on the verb the necessity of being performative: the action is executed entirely while the speech act lasts.

The nature of the quoted speech

• 'X,' she asked: X a request. ♦ 'X,' she complained: X is a complaint. ♦ 'X,' he exhorted: X is an exhortation. ♦ 'X,' he warned: X is a warning. ♦ 'X,' he promises: X is a promise. ♦ 'X,' he ordered: X is an order. ♦ 'X,' he answered: X is an answer.

• 'X,' she wondered: is X a spell of wondering? No, it is the subject of it, the wondering is not explicited in X as it is about X. Can this spell be considered to be confined to the time of the speech act? Hardly so; "wonder", on the basis of its definition is a dynamic verb (to think about sth and decide) but it seems to be perceived ambiguously as it is considered to be a stative verb, also as it is considered to be used in the progressive only exceptionally.

"The uttering of a performative is, or is part of, the doing of a certain kind of action. (Performative utterance) "
It follows that all performative verbs are first of all dynamic.

"Ponder", while it is a dynamic verb, is not performative: it does not characterize the quoted speech as an act of pondering and suffers from the same defect "wonder" does. Moreover, pondering is not the type of action that can be carried out in the quasi timeless span of such a short utterance, whether on the level of an inner dialogue or otherwise.
As to "muse" and "think", the same goes for them unless they can be taken as connoting a manner of saying (saying in a contemplative mood, saying silently something to oneself), and they are then verbs that fits in the first kind; the definitions in OALD does support this possibility in the case of "muse (2)"; it is not quite so for "think (3)" as its definition makes of it a stative verb; nevertheless I did put it in the first class.

As I plowed through the literature I couldn't pinpoint any reference to the effect of this specific licensing, but I believe that the evidence gathered tends greatly towards the non-licensing, although in a deductive manner.