Hint: what is your on-board clock doing as you approach the SOL, WRT a clock on a "stationary" frame of reference?
(think about the answer as it would affect the fact that acceleration is the derivative of velocity WRT time)
The clock is keeping perfect time, so far as the people on the ship are concerned. But the time dilation is real, as they would learn if they could return home. Similarly, I suspect (but don't know) that their mass increase is real too.