To my mind the fact that the name Grenapêche® wouldn’t be derived from grenadine shouldn’t be ruled out.

1- Ingredients
The fact that there is no grenadine (pomegranate) in the list of ingredients doesn’t stand. Grenadine is a soft drink made of grenadine syrup diluted in water. It’s long since grenadine syrup doesn’t contain the slightest scrap of pomegranate¹, except for some very expensive luxury brands or the do-it-yourself product. If any fruit used at all² in the off the shelf product they are : framboises (raspberries), cassis (blackcurants), groseilles (redcurrants), sureau (elderberries), fraises (strawberries), citron (lemon), and some grenadine syrups and use artificial food colouring as well.

So if we look at this Grenapêche® drink nothing in its ingredients prevents it from referring to grenadine, and on top of that it’s got peaches.
Hence the play on words: grenadine + pêche → grenapêche.

2 – Colour
Grenapêche® is of a "slightly cloudy orange colour" which is given by the paprika in it, and this is consistent with the colour of grenadine that can vary from dark red to a light pinkish colour.

3 – Hinting at "country"
We are now going to look into the part where the OP is trying to find a trace of "country style" in the French denomination of the product. To my mind "country style" could refer to something that is old fashioned, i.e. that is not modern/contemporary, and makes us think of our childhood. And that’s exactly what grenadine does to someone who grew up in France in the middle of the previous century and for whom sirop de grenadine is their madeleine de Proust.

Quoting from a 2019 magazine:

Elle nous met l’eau à la bouche. Pause écarlate, sirotée entre deux parties de chat perché, la grenadine a ce goût de l’enfance³. Si nos souvenirs sont authentiques, la boisson n’a de grenade que le nom. Dans le sirop, il n’y a, en effet, aucune trace de cette grosse baie rose à pulpe …

4- Subsidiary remark
The Canadian branch of this company seems to like playing on words when transposing the name of their products into French: their Sleepytime tea® becomes Tranquillithé® (tranquillité decomposed as tranquille+thé). Not up to me to judge…

^{1 "… l’une des plus belles arnaques de l’industrie agro-alimentaire de ces dernières années". Comment faire du sirop de grenadine maison
2 According to this wikipedia article in the US one can find grenadine without a single name of fruit in the list of ingredients.
3 My emphasis.}