My Latin is poor, so this is only guesswork too long to suggest in comments. Hopefully someone more knowledgeable can confirm my guesses here or correct my errors.
Elsewhere in similar historical records, obtinuit seems to mean "took office", as in chronologies of who served as bishop or archdeacon at a certain place, etc. For example, Prioratum obtinuit anno 1478: he became Prior in 1478. The verb obtineo suggests receiving something that already exists, and indeed in the small sampling of these chronologies that I looked at, this verb seems to appear often in sequences, describing the continuity of a single office as a succession of different people hold it. In many instances, obtinuit combines with ab to indicate who appointed the person to the office; e.g. the office was "obtained from" the Pope. I've also seen tradidit ("handed on").
If that is correct, then imperium obtinuit means "obtained the realm", that is, took over as ruler of all the kingdoms—became a sort of emperor. Some cursory googling suggests that this is indeed a common way of putting it. If so, then ducatum is not the object of obtinuit.
Many of the names of the medieval offices that go with obtinuit seem to end in -atum, which I'm guessing is cognate with English -ate in the same contexts. For example, Archidiaconatum obtinuit = he became archdeacon. Decanatum obtinuit = he became dean, though I'm less sure of this one. Note that in English we usually say "became archdeacon", but the Latin literally translated is "obtained the Archdeaconate".
Following that analogy, in this context ducatum might specifically mean "the Dukeship" or "a Dukedom" rather than leadership in general. See here for other instances of ducatum in similar records.
So, paring down the sentence, this:
Quartus Reduald rex Orientalium Anglorum, qui etiam uiuente Aedilbercto eidem suae genti ducatum praebebat, [imperium] obtinuit.
- Rædwald, king of the East Angles, became the fourth to hold rule over all the Anglo-Saxon kingdoms. Even while the previous such ruler, Æthelberht, was still alive, Rædwald granted a dukedom (or granted independence) to his (Æthelberht's) people.
A problem with this translation is that suae is reflexive and Rædwald is the subject. Possibly, though, Bede intended the apparently redundant eidem suae genti (to his own same people) to relocate the referent of suae to Æthelberht. This would agree with etiam, since the passage would point out something unexpected.
The big question is: did Æthelberht lose the imperium while he was still alive? If so, then the sentence by Bede would mean that Rædwald took the imperium while Æthelberht was still alive and gave the Kentish people some kind of official status as an autonomous dukedom. This chronology says that when Rædwald took the imperium, he did not control Kent, viewing it instead as a peer. So perhaps in this sense, he granted Kent ducatum, meaning autonomy, self-leadership (only a guess of mine).
If suae is Rædwald, then in order to make sense, you'd probably have to interpret ducatum as leadership, as in Colgrave's interpretation, with the somewhat hazy implication that the East Angles were moving up in the world during Æthelberht's lifetime:
- Rædwald, king of the East Angles, became the fourth to hold rule over all the Anglo-Saxon kingdoms. Even while the previous such ruler, Æthelberht, was still alive, Rædwald provided leadership to his (own) people.
By the way, notice that Æthelberht is called rex Cantuariorum, not rex Cantiæ, that is, king of the Kentish People, not king of Kent. This seems to agree with genti. However, the same applies to Orientalium Anglorum, so this doesn't help resolve the ambiguity.
Finally, note that praebebat is imperfect. My first translation above would fit a lot better with the perfect tense. The imperfect tense suggests the second reading, but I'm not aware of praebere being used in that sense. (That sense might have plenty of currency; I'm just not familiar enough with the literature to know. Can anyone here point out an interesting source?) So, for now, I'm favoring the first translation: Rædwald did not conquer Kent, even as he assumed the imperium from Æthelberht.
Why so ambiguous?
As for why the passage is so ambiguous, there seem to be several different ways to hook up the grammar and sense, none of which is fully satisfactory. Simon Keynes goes over several interpretations in "Rædwald the Bretwalda". It seems strange to say that Rædwald "supplied leadership" to his own people; isn't that what kings normally do? Why etiam, then, which suggests something beyond expectation? It might make sense to say that Rædwald offered Æthelberht leadership of the East Angles, but that would be ducatum suae gentis. Ducatum seems like it's there to contrast with imperium: it's a smaller sort of imperium. "Rædwald was leading his people's advancement during Æthelberht's imperium" (Colgrave's interpretation) violates common usage of praebebat. My translation #1 (originally proposed by Hanna Vollrath-Reichelt in 1971) seems to violate common usage of suae.
The trouble might actually result not from the grammar as much as from the ambiguous nature of the imperium itself. If the imperium is something vague, like a tendency on the part of leaders on the Continent to write to you when they want to address all the Anglo-Saxons through one person, but does not imply military supremacy or recognition at home, then it won't agree with modern expectations about hierarchical relations among leaders. Æthelberht might be bretwalda but at the same time he might sometimes be subservient to Rædwald; the situation changes continually as battles are won and alliances shift.
The imperium also violates the expectation that a ruler is ruler for life and that these rulers were consecutive with no long gaps between them. For example, Ceawlin's Wikipedia page says there was an approximately 50–year gap between Ælle and Ceawlin, and mentions that even during his bretwaldaship, Ceawlin often lost control of some of the kingdoms. The chronology mentioned above suggests that Æthelberht's failure to broker an agreement between the Roman Church and the Celtic Church around 603 damaged his credibility as bretwalda. Rædwald became bretwalda in 616, so there may have been another long gap.
The passage might even be Bede's way of saying that holding the Anglo-Saxon imperium doesn't necessarily mean that you were the top dog always and everywhere. And then again, it might just be a transcription error, as suggested by Henry Hallam here.