Consider, if you will, the phonetic term labialised. A labialised consonant, in my book, is one which has lip rounding/protrusion simultaneous with an articulation which has a greater degree of stricture — this means, in effect, a plosive, affricate, fricative, lateral or nasal. So, in English for example, the initial consonants of two, chew, soup, loose, noose etc. are likely to be labialised because of co-articulation with the following rounded vowel (if indeed the vowel has lip-rounding, which is becoming less and less the case these days). A phonetic transcription would record this labialisation as [tʷ sʷ nʷ] etc.
Now please consider words like ruse, rude, roof. Surely the same sort of thing is going on, isn’t it? Lip rounding is being anticipated during the articulation of the [ɹ]. So we have a labialised voiced postalveolar approximant and we can write [ɹʷ]. Er…just a mo. Is the stricture of [ɹ] more radical than that of the lip-rounding? Well, in most accounts of primary/secondary articulation you will find that the answer is no. What we have in the words listed above should really be classified as a double articulation — a labial-postalveolar approximant to be precise, just as [w] is classified as a labial-velar approximant and [ɥ] as a labial-palatal approximant. We don’t have a symbol for a labial-postalveolar approximant. May i suggest….

Or maybe I should get out more…
You speak sooth. Of course it’s only an allophone, but so are the w and ɥ of your analogy in French, for example, and a whole lot more other languages than people like to think (not that they do like to think). I would prefer [ɹ̫] if you can see that, but God knows what the latest pronouncements of the IPA are on how that compares with ɹʷ.
If you want a symbol for a labial-postalveolar approximant, wouldn’t you rather have a dedicated one like the said w and ɥ? Obviously that symbol is Jack W Lewis’s beloved wynn. ƿ looks a bit like an r and is a w! Alert the IPA!
Would [w͡ɹ] actually represent a labial-postalveolar-velar approximant, i.e. a triple articulation? This actually shows up the fact that there is no IPA symbol for a bilabial approximant, which is perhaps a more fundamental omission (I use [β̞]). As many English rs are labialised and pharyngealised, we could be having lots of fun with diacritics. I wonder if sometimes we might be dealing with a postalveolarised bilabial approximant, which might be a prime candidate for becoming labiodental (with or without a secondary articulation)…
Presumably this also relates to the differing degrees of rounding in [s] and [ʃ], and how they relate to rounding inherited from an adjacent vowel (“see”, “sue”, “she”, “shoe”, etc). Maybe words/symbols like labialisation/[ʷ] just don’t give us enough detail sometimes.
Anyway, if this sort of stuff is the result of not getting out enough, long may we all stay in!
Paul,
Yes, I agree about the possible “triplitude” of my suggested symbol, and the lack of a bilabial approximant symbol is certainly a problem. Thinking about it, we need more than one symbol really. Your [β̞] is fine, but it suggests (to me at least) a segment without lip-rounding or protrusion. We need one for lip action without accompanying back of tongue raising. At the moment, I have no suggestion for this.
I don’t much like the term postalveolarised bilabial approximant, however. In fact, the concept of X-ised approximants (if you see what I mean) makes little sense if you are dealing with a secondary vs primary classification system based on 3 degrees of stricture: closure ~ fricative stricture ~ approximant stricture.
Can we go out occasionally, please?
John,
Point taken about my made-up-on-the-spot term “postalveolarised bilabial approximant”.
Now we’re heading for [β̞ʷ]!
[ɵɹ] with a tie bar might be another solution.