The OP wants to find whether 1 Kings contain the following hints within the narrative:
Does the author intend us to understand that this oath is fabricated, such that Nathan and Bathsheba are attempting to trick David? If so, is this making some sort of statement about Solomon's rule? Or, is the author leaving the fact of the oath purposefully ambiguous to the reader? And if so, what is the purpose of doing so? Or is the reader simply meant to take it as face value that David made such an oath?
I found a 2014 journal article Reconsidering the Role of Deception in Solomon's Ascent to the Throne by Matthew Newkirk, published in the Journal of Evangelical Theological Society Vol 57 No. 4 reviewing arguments FOR and AGAINST deception, and offered his conclusion which is AGAINST deception.
The purpose of the this article is
to reconsider this prevalent view that Nathan
and Bathsheba deceived David into naming Solomon king. To do this, this article will proceed in three parts. First, I will briefly summarize the episode as found in 1 Kings 1. Second, I will consider the arguments advanced by scholars that David was deceived into naming Solomon king and seek to show that they are unpersuasive. Third, I will provide positive argumentation against deception in this passage
by highlighting evidence in the text suggesting that Adonijah is depicted as seditious and that Solomon is depicted as the rightful heir to the throne.
The arguments AGAINST is based on hints in the Deuteronomistic text (which includes 1 & 2 Samuels and 1 & 2 Kings) that characterize Adonijah as "seditious and therefore attempting to usurp the throne" versus hints in the text which portrays Solomon "as the the rightful king" plus of course, the witness of Chronicles.
Some excerpts of the hints in the Deuteronomistic text:
First, Solomon alone of David’s sons was not invited to Adonijah’s
feast at En Rogel (vv. 1:9b–10). Many attribute this exclusion to two rival factions
existing in the court—one Hebronite and one Jerusalemite—and thus Adonijah
simply did not invite his rival. However, since Solomon was not the only son of
David born in Jerusalem (see 2 Sam 5:13–16), this does not explain why Solomon
alone of David’s sons was excluded. Gwilym Jones claims that nothing in the text
indicates that Adonijah knew about an oath to Solomon, but Adonijah’s exclusion
of Solomon alone may be that very evidence indicating that he was aware of such
an oath. If Nathan knew about the oath, it is plausible that others within the court
could have known about it as well.
The second hint comes from Adonijah when he later tells Bathsheba, “You
know that the kingdom was mine and all Israel looked to me to be king, but the
kingdom turned and went to my brother, because it was his from YHWH (
היתה לו כי מיהוה
)” (1 Kgs 2:15). Here Adonijah himself admits that Solomon had a divine
right to the throne. Halpern acknowledges the significance of this statement for
legitimatizing Solomon’s accession, yet because he insists that the text is attempting
to cover up the fact that Adonijah was the true heir apparent, he concludes that
“this admission is the work of the apologist.” As was the case with Ishida’s observations
noted above, this argument is wholly conjectural and supports the view that
the final form of the text is portraying Solomon as the rightful successor.
Third, the only comment the narrator makes about Solomon before this episode
is that "YHWH loved him (ויהוה אהבו
)" (2 Sam 2:24). This notice singles out
Solomon early in the narrative as the recipient of divine favor, and therefore that
the kingship “was his from YHWH,” as Adonijah admitted, is plausible. K. L. Noll
goes to great lengths to argue that the verb אהב here does not imply that YHWH
chose Solomon as David’s successor. Yet it is not necessary to conclude that אהב in itself designates Solomon as successor; it simply shows that from his birth Solomon
is characterized as in a special, favored relationship with YHWH, which lends
plausibility to the idea that he was YHWH’s choice.
The last hint of Solomon’s right to the throne is the onomastic puns in this
chapter. As Moshe Garsiel insightfully observes, puns on the names of the two
brothers’ mothers correspond to the tactics used by the two men in their attempts
for the throne. On the one hand, although the word חג (“feast”) does not occur
in this chapter, Adonijah’s sacrificial gathering is described in feast-like terms.
This corresponds to Adonijah’s mother’s name, חגית (Haggith), which is from the
root חג, with whom he is associated three times: “Adonijah the son of Haggith”
(1:5, 11; 2:13). On the other hand, in these chapters Bathsheba is twice referred to
as “the mother of Solomon” (1:11; 2:13), and it is twice stated that David “swore”
(שׁבע) that Solomon would be king (1:13, 17). This corresponds to her name,
בתשׁבע (“daughter of an oath”). If the narrator is using these puns to “enrich and
intensify the plot,” as Garsiel suggests, it could imply that just as Adonijah actually
had a feast by which he attempted to attain the throne, so did Solomon actually
have an oath by which he successfully attained the throne. If this is the case, these
puns on the names of the two rivals’ mothers serve as further literary indicators
supporting the veracity of the oath and thus the validity of Solomon’s kingship.
In this article I have considered the arguments that Nathan and Bathsheba
deceived David into naming Solomon king and concluded that they are unpersuasive.
Nothing in the text suggests that David should be viewed as senile or easily manipulated. Rather, the narratological emphasis on David’s old age is connected
to his political ignorance. In conjunction with David’s history of failing to correct
Adonijah, this explains why Nathan and Bathsheba orchestrated their appeal for
David to actualize his oath that Solomon succeed him as king. I then argued that
the textual evidence suggests that Adonijah is depicted as seditious, that subtleties
in the passage suggest that Solomon was the rightful heir to the throne, and that
the book of Chronicles reinforces this interpretation by portraying David as knowledgeable
that Solomon was YHWH’s choice to succeed him. Rather than being the
result of human duplicity in the royal court, Solomon’s ascent to Israel’s throne was
the fulfillment of YHWH’s word to David in line with his covenant promises.