According to this description of the English phonotactics, the schwa /ə/ doesn't occur in stressed syllables. But Cambridge Dictionary Onlines, Oxford Advanced Learner's Dictionary and Longman Pronunciation Dictionary say otherwise: the word "because" has been variously transcribed as /bɪˈkəz/ or /bəˈkəz/. So in which source should I believe? Or is there any error in transcription going on here? Or is "because" the only exception? And are there others?
I think the source of your confusion is that different people use different sets of symbols to represent the same pronunciations in English. Phonemic notation for English is fairly standardized, but it isn't absolutely uniform: for example, another area of variance is the notation of syllabic resonants (do we write the last syllables of button chasm bottle butter as /n̩ m̩ l̩ r̩/ or /ən əm əl ər/, or perhaps even /ᵊn ᵊm ᵊl ᵊr/?) and the notation of the "r" consonant sound (do we go with the simpler /r/ or the possibly more phonetically accurate /ɹ/?). In general, the difference between these is completely meaningless. Check the pronunciation guide of whatever resource you are using to learn what phonemic value some particular symbol represents.
When the schwa symbol <ə> is used in a stressed syllable, it almost certainly represents the STRUT vowel, even though this is standardly represented by /ʌ/. The "stressed schwa" notation for this vowel is probably most common in ad-hoc dictionary phonemic notation systems (I'd guess because it is more familiar to the general public than <ʌ>). Usually, notation systems that represent STRUT with <ə> will also represent NURSE with <ər>, <əɹ>, or <ɚ> (instead of <ɜr>, <ɜɹ>, or <ɝ>). It might not be entirely phonetically accurate, but neither is [ʌ] for many speakers (who may have a phonetic value more like [ɐ] or even [ə] instead).
This notation is possible without much, if any, ambiguity because schwa and STRUT are minimally contrastive at best, and actually in complementary distribution for many speakers. (See Accents of English by John C. Wells (who is also the author of the Longman Pronunciation Dictionary) for a description of this.)
The use of this notation in the Merriam-Webster dictionary was discussed at the following usingenglish.com forum thread: ə(unstressed) vs ʌ(stressed). The dictionaries you list evidently follow the same practice.
However, an earlier part of the Wikipedia article you referred to ("Vowels") defines the STRUT vowel as /ʌ/.
So the two sources simply use different definitions. The statement in the Wikipedia article is true, given the notation used there: it isn't true for all possible, or even for all existing phonetic notation systems used to transcribe English.
Note: I may be biased towards identifying /ə/ with /ʌ/ here because I do speak a dialect where "because" and "was" rhyme with "buzz," and "what" rhymes with "cut."
Peter Shor's explanation that it represents an unstressed weak form is also possible. It seems particularly likely for the Cambridge online dictionary; the evidence for it is that STRUT is usually represented by the symbol <ʌ> (I can't find any (other) word where /ə/ represents STRUT) , and for "but," they distinguish "strong /bʌt/ weak /bət/." The transcription in general doesn't always seem to be consistent; in the UK dictionary, "muzzle" is transcribed as /ˈmʌz.l̩/ but "guzzle" is transcribed as /ˈɡʌz.əl/.
In the pronunciation /bəˈkəz/, neither of the syllables is truly stressed.
There are several function words in English which have weak forms and strong forms, depending on whether the word is stressed in the sentence. For example, the words but, just, could, should are pronounced with /ʌ/, /ʊ/ when they are stressed, and /ə/ when they are unstressed, which is usually the case.
I believe that the pronunciation /bəˈkəz/ is a weak form. We can have a schwa in both syllables of /bəˈkəz/ because in the weak form, neither vowel receives much stress.
In fact, I have the weak form /bɪkəz/ for because, which I use when neither syllable receives stress. For example, if I was saying "because John is ..." in informal speech, I would probably use /bɪkəz/. On the other hand, if I was saying "because of the ...", I would definitely use /bɪˈkɔz/, simply /bɪkəz/ English doesn't like long strings of unstressed syllables. For me, the /ə/ and /ʌ/ are vowels with different qualities, and I don't believe I ever say /bɪˈkʌz/, with or without stess on the /ʌ/.
Why do the dictionaries put a stress mark on the second syllable in this pronunciation? You'll have to ask them, but I assume they have a rule that two-syllable words need stress marks. Why don't they mark it as a weak form, the way they do for but? You'll have to ask them that, too.
Other exceptions would have to be two-syllable function words with weak forms. I can't think of any off-hand, but there probably are some.
In my opinion, and I think also in the opinion of many other American linguists, there is no stressed sound [ə] in standard American English. This sound occurs only as a result of the phonological process of vowel reduction in certain unstressed syllables: non-high lax unstressed vowels become schwa.
There is no phoneme schwa in English, so the notation /ə/, taking the slashes to signify phonemic forms, is not interpretable for English. It doesn't make sense.
However, even among American linguists, there is probably considerable disagreement with what I have just said, and there seem to be some very different ideas about schwa, notation, and phonemics in general in the uk, so I'm afraid you can't count on what I have said to be helpful as a guide to understand what people say or write about schwa. Sorry.