It was the Buddha who spoke the words of the DN, but it was compiled, collated, arranged and rearranged by the Sangha centuries after his passing. The DN was not fabricated. Also the Buddha certainly never said, "Today I'm going to speak Digha Nikaya 1".
The Sangha may have rearranged the suttas into the sequence of the DN today with intention to define the difference between the Buddha's teachings and other religions (in DN 1), the role of the monks (in DN 2), the role of the lay followers (in DN 31), the continuation of Buddhism after the Buddha passes away (in DN 16) etc. There might have been some specific intention behind this arrangement, perhaps to create a constitution of sorts.
OP: Is there evidence (such as contradictions with other suttas) suggesting the Buddha did not speak the Digha Nikaya?
There is no evidence of contradiction between the nikayas.
OP: In there any evidence in the suttas where the Buddha said he would teach a different modified Dhamma ("propaganda") merely for the
purpose of converting Brahmins & other outsiders?
No. The Buddha taught the same thing to monks, lay followers and outsiders, according to SN 42.7:
“To me, the monks and nuns are like the good field. I teach them the
Dhamma that’s good in the beginning, good in the middle, and good in
the end, meaningful and well-phrased. And I reveal a spiritual
practice that’s entirely full and pure. Why is that? Because they live
with me as their island, protection, shelter, and refuge.
To me, the
laymen and laywomen are like the average field. I also teach them the
Dhamma that’s good in the beginning, good in the middle, and good in
the end, meaningful and well-phrased. And I reveal a spiritual
practice that’s entirely full and pure. Why is that? Because they live
with me as their island, protection, shelter, and refuge.
To me, the
ascetics, brahmins, and wanderers who follow other paths are like the
poor field, the bad ground of sand and salt. I also teach them the
Dhamma that’s good in the beginning, good in the middle, and good in
the end, meaningful and well-phrased. And I reveal a spiritual
practice that’s entirely full and pure. Why is that? Hopefully they
might understand even a single sentence, which would be for their
lasting welfare and happiness.
So, he did not teach followers and non-followers differently..
OP: Is there any evidence in the suttas showing the Buddha was "flexible" in his teaching of Dhamma? Or do the suttas show the Buddha
wished his teachings to remain consistent & clearly representative of
what he taught?
WHAT he taught was the same, but HOW he taught it, could be different according to the Kesi Sutta (AN 4.111):
"Kesi, I train a tamable person [sometimes] with gentleness,
[sometimes] with harshness, [sometimes] with both gentleness &
harshness.
"In using gentleness, [I teach:] 'Such is good bodily conduct. Such is
the result of good bodily conduct. Such is good verbal conduct. Such
is the result of good verbal conduct. Such is good mental conduct.
Such is the result of good mental conduct. Such are the devas. Such
are human beings.'
"In using harshness, [I teach:] 'Such is bodily misconduct. Such is
the result of bodily misconduct. Such is verbal misconduct. Such is
the result of verbal misconduct. Such is mental misconduct. Such is
the result of mental misconduct. Such is hell. Such is the animal
womb. Such the realm of the hungry shades.'
"In using gentleness & harshness, [I teach:] 'Such is good bodily
conduct. Such is the result of good bodily conduct. Such is bodily
misconduct. Such is the result of bodily misconduct. Such is good
verbal conduct. Such is the result of good verbal conduct. Such is
verbal misconduct. Such is the result of verbal misconduct. Such is
good mental conduct. Such is the result of good mental conduct. Such
is mental misconduct. Such is the result of mental misconduct. Such
are the devas. Such are human beings. Such is hell. Such is the animal
womb. Such the realm of the hungry shades.'"
"And if a tamable person doesn't submit either to a mild training or
to a harsh training or to a mild & harsh training, what do you do?"
"If a tamable person doesn't submit either to a mild training or to a
harsh training or to a mild & harsh training, then I kill him, Kesi."
"But it's not proper for our Blessed One to take life! And yet the
Blessed One just said, 'I kill him, Kesi.'"
"It is true, Kesi, that it's not proper for a Tathagata to take life.
But if a tamable person doesn't submit either to a mild training or to
a harsh training or to a mild & harsh training, then the Tathagata
doesn't regard him as being worth speaking to or admonishing. His
knowledgeable fellows in the holy life don't regard him as being worth
speaking to or admonishing. This is what it means to be totally
destroyed in the Doctrine & Discipline, when the Tathagata doesn't
regard one as being worth speaking to or admonishing, and one's
knowledgeable fellows in the holy life don't regard one as being worth
speaking to or admonishing."