My first instinct was to answer that of course they are related. Both words are of biblical origin, and milk has a high fat content (particularly as was consumed in ancient times). And, I thought, a parallel could be made with shuman שומן - "fat" (the kind permitted to eat) and shamenet שמנת - "cream".
But if there's one thing years of writing on Balashon has taught me, is that my first instinct is often wrong. And it certainly was this time.
Sometime when I look at etymologies of Hebrew words, I'm comfortable looking at pre-modern sources. The problem with doing that in cases like this, is that the temptation to connect such similar words is great, and without the assistance of modern linguistics, it was nearly impossible for earlier scholars to get to the real origins of the words.
So in this case, I went straight to Klein (made much easier by Sefaria's digitized edition of his book).
Here is his entry for chalav:
חָלָב m.n. milk. [Related to Aram. חֲלַב, Syr. חַלְבָּא, Ugar. ḥlb, Arab. ḥalab, ḥalib, Ethiop. ḥalīb (= milk). Akka. ḥalābu (= to milk).]
And here is his entry for chelev:
חֵֽלֶב m.n. fat, grease. [Related to Phoen. חלב, Syr. חֶלְבָּא, Arab. ḥilb (= midriff). The orig. meaning of these words was perhaps ‘fat of the midriff’.) ]
The two aren't related, and I couldn't find any modern source that did connect the two.
But it turns out I wasn't only wrong about that. I thought that shuman and shamenet were also biblical words. Nope. Shuman was introduced during the Talmudic period (and is related to the biblical word for oil, shemen שמן). Shamenet is actually very modern word, only being coined in 1933. It replaced Ben Yehuda's word for cream - zivda זבדה (based on the Arabic zubda - "butter, cream".) Ben Yehuda writes that he chose that word, because the biblical word for cream - chemah חמאה - had become in his time used for the product of churning cream - i.e. butter - a new word was needed for cream.
And while shamenet is certainly based on the root שמן (connecting it to shemen and shuman), that wasn't why it was chosen. Rather, there was already a common Yiddish word - shmant - meaning "cream". And shmant doesn't have any Hebrew cognates at all. It's directly related to the German schmand (and therefore likely a distant cousin of the English word "smooth".)