Does English allow /eɪʃ/ in the end of a syllable (in the same syllable)?

I'll start by saying this is not a very answerable question.

• For one thing, it's far from obvious that rules about what "English allows" are 100% knowable. Some sequences of sounds exist in only one or a handful of words; e.g. the Wikipedia page List of English words without rhymes gives gouge, scarce, ninth as the only words ending in the rimes /ˈ-ɛərs/, /ˈ-aʊdʒ/, /ˈ-aɪnθ/.

  This suggests that there might be many "accidental gaps" in attested sequences of sounds: sequences where it is just chance that no word, rather than one or more words, contains the sequence. I believe the existence of accidental gaps in syllable rimes is widely assumed by linguists, although people may differ in what they consider to fall in this category.

• The other issue is that there is no perfect definition of what it means to be an English word, so you'll run into problems with deciding things like whether Kshatriya, which has an entry in Merriam-Webster with a pronunciation transcribed as starting with /kʃ/, counts as proof that "English allows" /kʃ/ at the start of a syllable, or iamb, which has an entry showing a pronunciation ending in /mb/, counts as proof that English allows /mb/ at the end of a syllable.

My view is that it makes more sense to say that English does allow /eɪʃ/ at the end of syllables than to say that it doesn't.

• I'm an English speaker, and I don't find it difficult or particularly remarkable to pronounce a syllable ending in /eɪʃ/

• There is a name Naish that seems to be pronounced at least sometimes with /eɪʃ/: listen to these Youglish results for Naish

A comment on your description "vowels before digraphs are usually short and not long." As you point out, this is a spelling-based rule, so it isn't directly relevant to phonotactic rules, which are conceived of by linguists as rules about phonemes. In any case, there are a non-negligible amount of exceptions to this generalization, such as truth, Ruth, faith, speech.