I think the distinction sidegrid was making is not actually about pronunciation but about the two uses of have as an auxiliary verb and as a verb indicating possession.

https://en.wiktionary.org/wiki/have#Verb

My impression is that standard American English only contracts the auxiliary verb ("I've biked up Mount Tamalpais") and not the possession verb (?"I've a pair of prescription sunglasses"). Hence "I've been diagnosed with bronchitis" (auxiliary) but not ?"I've a case of bronchitis" (possession).

However, I think this rule is different in Commonwealth English, so we might just be witnessing a difference in English varieties.

As a Commonwealth English speaker, I am not aware of a rule preventing contraction of the possessive verb "to have." Maybe there is, and I'd be interested to learn about it.

Having said that, commonly I wouldn't say, "I've a case of bronchitis," rather just "I've bronchitis." It feels more natural to say "I have a case of bronchitis." However, this may just be personal preference.

I'm now pretty sure that the rule is specific to American English.