(kharpare śari vā visargalopo vaktavyaḥ)
Let's translate this piece by piece to understand its logic:
śari: "when a śar consonant follows (the visarga)". (śar = श्, ष्, स्).
kharpare: "when a khar consonant is following that". This is a compound word: khar (unvoiced consonants) + pare (following). So, this means "when the śar is followed by a khar".
vā: "optionally".
visarga-lopaḥ: "the elision (dropping) of the visarga".
vaktavyaḥ: "should be stated".
So, the rule is not simply "drop the visarga before a śar". It is a two-level condition:
Optionally drop the visarga IF:
It is immediately followed by a śar consonant (श्, ष्, स्), AND
That śar consonant is itself immediately followed by a khar consonant (unvoiced stops and sibilants: क्, ख्, च्, छ्, ट्, ठ्, त्, थ्, प्, फ्, श्, ष्, स्).
Applying this to तिसृभि: + स्तुवन्ति:We have the visarga (:) from तिसृभि:.
Is it followed by a śar? Yes, the स् in स्तुवन्ति is a śar consonant.
Is that स् (śar) followed by a khar? Yes, the त् in स्तुवन्ति is a khar consonant.
Since both conditions are met, this vārttika applies, and we are given the third option of dropping the visarga, resulting in तिसृभि स्तुवन्ति.
Why the khar Condition is Crucial (The "Anuvṛtti" Logic)The "anuvṛtti" (continuation/inheritance of context) you mention is conceptual. The entire section of the Aṣṭādhyāyī is governed by contexts. The primary rule 8.3.34 visarjanīyasya saḥ introduces the context of khari (when a khar follows). The vārttika builds on this by creating a more specific context.
The kharpare condition is there to restrict the application of this visarga-dropping rule. It prevents it from applying too broadly.
Consider a counter-example where the rule does not apply:
रामः शेते (rāmaḥ + śete)
We have a visarga (:).
Is it followed by a śar? Yes, श् is a śar.
Is that श् (śar) followed by a khar? No. It is followed by the vowel ए, which is not a khar.
Because the kharpare condition fails, the vārttika for dropping the visarga cannot be used. You cannot say राम शेते. The only valid options here (from other rules) are रामः शेते and रामश्शेते.
In summary: The khar condition is not a background anuvṛtti in this case, but an explicit and essential part of the vārttika. Its purpose is to define a very specific phonetic environment (visarga + sibilant + unvoiced consonant) where the elision of the visarga becomes a valid third option.