Peace,
The only reason in my mind why the fat of animals has been singled out for Jews is because they had very stringent rules of what they could and could not eat. See verse 6:146 of the Quran and the Old Testament. That is why the Quran uses the distinguishing terms.
Please note that if the same logic is applied, the reverse argument is just as sound. For example, it can be argued that by the using the term 'khanzeer', the Quran has forbidden the whole animal, as if the Quran wanted to allow other parts such as 'fats' (shuhumahuma), their entrails (hawaya) etc, the Quran could have mentioned it, especially when it was forbidden before and mentioned in the Quran. But as the Quran didn't mention it, hence, the whole animal is intended in the prohibition.
I hope you can see that the reverse argument is also just as cogent if the same logic is applied. It arguably seems more plausible.
The reason I feel the 'lahmu' (flesh) of 'khinzir' (swine) has been used is that given the context, the verse is talking about 'eating' and mentioning the 'flesh' of the prohibited animal is in keeping with the flow of the verse and Arabic. If you one consults verse 56:21, where the flesh (lahmi) of fowls will be given to those in paradise as a similitude of that of on Earth, would this exclude the insides? Or is the 'lahmu' a representation of the whole animal?
'and the flesh (lahmi) of fowls of they desire' (56:21)
If here the whole fowl is intended (including the fat), then why would 'lahmu' with regards to khanzir (swine) not be considered as a representation of the whole animal?
I hope this helps, God willing.
Joseph.