Hi Fallingstar1971.
Very interesting & inteliigent question.
I would like to contribute something here.
The short answer is no.
But let me explain.
Fomalhaut quick facts.
Fomalhaut is at most 200 million years old (0.2 GYO, Sun 4.6 GYO), so the age is just less than the time since The Great Dying. Fomalhaut is approx 2.1 times the Sun's mass, 18 times more luminous than our Sun & is approx 1.8 times the Sun's diameter. Fomalhaut's is an A3 star yielding a 'surface' temperature of 8,300 Celsius, our Sun is a G2 star yielding a 'surface' temperature of 5,500 Celsius. Fomaulhaut rotates on it's axis once every 1 day & 2 hours (equatorial, like our Sun probably has differential rotation, the polar regions being slower) as against our Sun's 25 days 8 hours (equatorial).
Fomalhaut is thought to be part of the Castor moving group, which includes Castor (obviously & AKA Alpha Geminorum), Vega (Alpha Lyrae), Alderamin (Alpha Cephei), Zubenelgenubi (Alpha Librae), Zeta Leporis (central star of a gigantic asteroid belt), & Psi Velorum amongst other lesser known stars.
All of these stars are thought to have formed together within a volume of only a few light years of one another in the same nebula, then for several million years were part of the same open cluster (like the Pleiades M45), then since have scattered, but still share the same general orbit around the centre of the Milky Way.
Regarding the ignition of Fomalhaut & The Great Dying. Certainly not.
1). Fomalhaut is in a different orbit around the centre of the Milky Way than our Sun & is currently receding at approx 6.3 KM per second. Closest approach to our solar system has already occured.
2). 250 million years ago, Fomalhaut was only a protostar at best.
3). 250 million years ago, Fomalhaut would have been in a very different part of the Milky Way, much further away, possibly several 1,000s of light years away.
4). The Great Dying was almost certainly entirely of the Earth's own making. Enormous continent Pangea formed, creating a gigantic hot desert in it's interior. Increased global temperatures resulted from this, this alone would have lead to the extinction of most land dwelling species, & chemical reactions in the oceans lead to lower PH levels (increased acidity) killing off most marine life. There was also increased volcanism during this period, leading to increased toxicity in the atmosphere.
The KT extinction 65 MYA, was certainly two fold, comet impact & mass volcanism together.
Andrew Brown.