Here are the current results of my further research:
The pronunciation of Boadicea with antepenult stress is found at least as far back as 1813 (as mentioned in the original question, recorded in Coxe's New Critical Pronouncing Dictionary) and at least up until 1870 (recorded in Beeton's Dictionary of Universal Biography, alongside the penult-stressed variant). It's unclear to me if there was ever a strong consensus in its favor.
The pronunciation with stress on the penultimate e seems to have arisen no later than 1848 (recorded in Mile's American Mnemotechny), and was apparently firmly established (at least in prescriptive circles) as the standard pronunciation of Boadicea by the start of the 20th century, although by this point, some people had begun to feel that the spelling "Boadicea" was itself problematic.
I'm still hoping to learn more about the history of this name's pronunciation if I can.
I haven't found any explicit descriptions of the pronunciation of Boadicea from this century. In verse, it seems to scan as four syllables, stressed-unstressed-stressed-unstressed, but unfortunately it doesn't seem entirely clear to me whether that represents something like bo´a-dish´a with an elided penult syllable, or bo´di-se´a with an elided/contracted first or second syllable. However, I would guess it is the former.
"Britain", Part III of the poem Liberty, by James Thomson, in The Works of James Thomson, Volume Two (London 1757):
As when Caractacus to battle led
Silurian swains, and Boadicea taught
Her raging troops the miseries of slaves.
(Note that the last "i" in "Silurian" does not take up a syllable of the meter either)
Boadicea: A Tragedy, by Richard Glover (1791):
Her children, friends, and country ; then recal,
What once was Boadicea, fall'n how low
(Act III, p. 41) (There are a number of other examples from this play)
American Mnemotechny, Or, Art of Memory, by Pliny Miles, Third Edition (New York 1848) contains a transcription in a special alphabet devised by Andrew Comstock that's equivalent to "bōădĭsē´a" or IPA /boʊædɪˈsiːɑː/ (p. 251). Oddly enough, the name "Augeas" is transcribed on the same page with antepenult stress (as the equivalent of "au´jēăs" or IPA /ɔːdʒiæs/).
New Classical Lexicon of Biography, Mythology, and Geography, by Thomas Swinburne Carr (London, 1858):
Boadicēa, a queen of the Iceni, in Britain (p. 65)
A Dictionary of the English Language, by Joseph Emerson Worcester (Boston, 1860):
BEA, CEA, DEA.
Accent the Penultimate.
Rhobea, Colacea, Gylacea, Pharmacea, Anacea, Panacea, Sphecea, Boadicea, Laodicea, Micea, Stratonicea, Ericea, Lancea, Ladocea, Cymodocea, Dorcea, Lyrcea, Polydeucea, Lebadea, Medea, Diomedea, Midea, Brasidea, Budea.
Beeton's Dictionary of Universal Biography, Second Edition, by Samuel Orchart Beeton (London 1870):
BOADICEA, bo´-a-dis´-e-a, or bo-a-di-se´-a (p. 177)
Seven Thousand Words Often Mispronounced, by William Henry P. Phyfe (New York, 1889):
Boadicea (Queen)—bō-ăd-ǐs-ē´ȧ. (p. 104)
The Chautauquan: A Monthly Magazine. Volume 12. October 1890 to March 1891. "C. L. S. C. Notes on Required Readings. For October."
"Boadicea" [ba-ad-i-se´a] (p. 106)
A relevant entry with fairy extensive discussion occurs in Notes and Queries, Ninth Series, Volume I: January–June 1898:
BOADICEA (8th S. xii. 366, 497).–The question asked by C. C. B., how this name should be accented, is one I have often asked myself. The modern Welsh Fóeddawg is accented upon the penult, but must once have been accented upon the final (Foeddáwg), as is proved, among other things, by the presence in it of the diphthong aw, derived from an older o by the action of the stress; at any rate, I know of no other reason which could account for this diphthongization. But the really important thing is to find out which of the numerous spellings of this name is the most correct. Here our best authority is Prof. Rhys, who pronounces in favor of Bodícca or Boudícca, both of which forms actually occur in inscriptions. Camden's Voadíca or Boodícia and the other variants quoted by C. C. B. are all what Prof. Rhys calls the "gibberish of editors." It is noteworthy how the terminal -cca has bothered the copyists, who have turned it into -cia or -cea; and the pronunciation which we have all learnt in the classroom, and which has been blindly followed by Tennyson (Boadicéa), is therefore absurd in so far as the stress falls upon a totally imaginary vowel for which there is no warrant. On the whole, those orthographies which do not show this intrusive vowel ought to be preferred, such as Camden's Voadíca, mentioned above, or Bondúca, and I consider that these should be accented, as I have marked them, upon the last syllable but one.
James Platt, Jun.
(p. 94; Jan 29, 1898)
I found a poem from 1902 that seems to give the penult-stressed version of the name "Boadicea". The poem was published in Punch, or the London Charivari, July 2, 1902 (Volumes 122-123); it is called "BOUDICCA". It uses the name "Boadicea" in the same place in the meter as lines like "On the Embankment!", "Briefly victorious", "Centuries after", "Since your last advent", "Wasn't included", and "Footnote or something", which all have five syllables with a stress on the first and the penult syllable.
There are a number of prescriptive sources from a few years later that provide clear evidence of the penult stress being used often enough to be considered "correct" in the early 20th century:
A Manual of Pronunciation, by Otis Ashmore (Boston, 1904):
bō-ăd-ǐ-sē´ȧ (p. 52)
(I'm not sure if the use of unreduced "ăd" in the second syllable of the transcription indicates that Ashmore put secondary stress on this syllable rather than on the first, or if it's just a sign that he was one of the kind of orthoepists who believed vowel reduction should be avoided whenever it is at all possible to imagine using an unreduced vowel. He does explicitly transcribe secondary stress using a small acute accent in some other names, but for some reason not in this one.)
The Pronunciation of 10,000 Proper Names, by Mary Stuart Mackey and Maryette Goodwin Mackey (New York, 1909):
bō-ȧ-dǐ-sē´ȧ. (p. 43)
The New Century Book of Facts, by Carroll D. Wright (Springfield, Mass., 1909):
bō-ăd-ǐs-ē´ȧ. (p. 88)
Correct Pronunciation, by Julian Abernathy (New York, 1912):
Boadicea—bō´a-dǐ-sē´a. (p. 117)
A phonetic dictionary of the English language, by Hermann Michaelis and Daniel Jones (1913):
(comparison with the other entries reveals that Michaelis and Jones are using the acute before the stressed syllable, rather than after it; the lack of a length marker after the "i" seems to indicate a pronunciation ending in the sound of "sear"; i.e. rhyming with words like "deer" or "fear")