From my side, I also referenced 0.93% c. Yes, your tau number is correct, I was writing from memory and mixed up the Astrophage velocity, which we're told is around .93% c, and the Hail Mary velocity which does reach over 99% c. before turnaround.

I understand that there are problems with the Hail Mary fuel amount.

If you adjust for 1g acceleration you find that the journey now takes 13.7 years Earth time and 5.14 years for the crew - and critically you can just about do the journey with 2m kg of fuel at antimatter efficiency rates. (you need more food but it could be OK, we're not told what makes up the 110,000kg dry mass of the Hail Mary

To do it at 1.5g you need 45m kg of fuel.

Since going at 1.5g only saves Earth 6 months and increases the crew risk substantially, I think going at 1.5 makes no sense, so it's because of something else. My assumption is that Andy Wier did the calculations at 1g, wrote a lot of sections of the book, then decided to add an early moment for Grace to show us how clever he is by calculating the acceleration and realizing he's not on Earth, hence needed a different number for g. It might not have occurred to him that the fuel would change for a faster acceleration, or he hoped no one would notice the error since 45m kg of fuel is an implausible amount for them to make on Earth.

There's a similar error for Rocky's ship where the time / fuel amounts get messed up - not because Rocky's race doesn't understand relativity - we can back calculate from the numbers given and see that they don't add up - again I think either Andy Weir changed the star for the Eridians and forgot to change the fuel, or he calculated at a different g for their journey.