It’s definitely an Anglicism. I think it marks you as belonging to the Facebook generation. As a data point, I’m pre-Facebook and don’t use it. I’m pretty sure it didn’t exist until the mid- to late 2000s, and I don’t think it’s sufficiently widely adopted to consider it a common French word rather than a fad.

Unsurprisingly, French has many words to express antagonism and attacks. “Tu vas te faire taper [dessus]” is pretty generic, both in terms of meaning (literal hitting or verbal attacks) and in terms of register.

I fully agree with Gilles about basher usage in French.

Encore que tu n’as aucune envie de faire des jalouses. is dubious.

About what verb could be used without being such an Anglicism, I would suggest te faire chambrer.

Here is a rephrasing of your sentences to better match what might be used colloquially:

Sur Instagram, t’as posté une photo où t’es super belle et à tous les coups, tu vas te faire chambrer grave (or on va se foutre de toi) ! Pourtant, tu voulais pas faire de jalouses. (rires)

Written with "sh" it surely qualifies as an anglicism, just like "bashing" which got a peak of popularity 5-10 years ago in the French medias.

But the use of "bâcher" in a similar meaning (to make a vexing remark) predates this new usage, it was widely used during my collège years in the early 90’s, as well as the use of "bâche" for a vexing remark.

That said, according to Wiktionary, "bâcher" in this sense appeared only in the 80’s, and that’s newer than I thought. There is no evidence of an English origin, though, but I suspect "basher" rooted so well in French because of the similarities in pronunciation and meaning.