I'm always a bit leary of Amazon descriptions, but equally machine auger bits have always seemed to start at 6mm in the past. Hand auger bits, designed to be used in a bit brace, used to start at 3/16in or 4.8mm, but they are increasingly difficult to get hold of and TBH they aren't designed to be used in a power tool
You are right about the pilot hole needing to be a bit smaller, but if it's two small with such a long screw the screw can wind-up (twist) and snap off like a carrot, especially with softer screw materials like stainless steel or brass. That is certainly a very long screw! TBH I didn't think anyone did stainless steel screws of that size. Why so long?
If the screw thread extends for only a short distance in that 150mm (e.g. 40mm) you might need a smaller pilot, but for 100mm of thread length into timber 6mm should suffice
Anyway, try this
£3.29 clearance offer from Toolstation - 6mm diameter, 235mm long, working length 160mm. It's a Heller brand. I have a few Heller auger bits in my auger bit roll and they're reasonable quality (i.e straight, reasonably sharp, more or less the size they say they are and they can be re-sharpened). Alternatively, on eBay, there's this
Bosch 6mm auger bit, 235mm long x 160mm working length. Cheapest for this item on
Amazon is nearly 3 times the price, at £12.79!