नमस्ते श्रीमन् मुत्तुकृष्ण-रामानुजं-वर्य !
I am left wondering why you think that in "भवान् + चरति becoming भवाँश्चरति / भवांश्चरति", there is some step of "न् changes to र्" (?!)
What rule we are looking at is नश्छव्यप्रशान् (8'3'7). Here नश्छव्यप्रशान् = नः छवि अप्रशान्
Let us focus on "नः छवि" leaving aside "अप्रशान्", because that is an exception.
"नः छवि" means something should become of न् when followed by छव्.
In "भवान् + चरति" we do have न् followed by छव्, since च belongs to प्रत्याहार छव् (= छ् ठ् थ् च् ट् त्).
So "something should become of न्". What is that something, that should happen of न् ?
By two सूत्र-s, previous to नश्छव्यप्रशान् (8'3'7), we get अनुवृत्ति-s of अनुनासिकस्य अनुस्वारः and सुट् (= आगम of स्).
In the case on hand, we have अनुनासिक न् which should become अनुस्वार and we should also have सुट् (= आगम of स्).
By that we get भवान् + चरति = (भवां or भवाँ) + स् + चरति
Now at "स् + चरति" we have the case, which is eligible to apply स्तोः श्चुना श्चुः.(8'4'40) Hence स् would become श्.
All in all भवान् + चरति = (भवां or भवाँ) + स् + चरति = (भवां or भवाँ) + श् + चरति = भवांश्चरति or भवाँश्चरति