Accelerating at Earth's gravitational acceleration around 9.81m/s^2 you approach light speed of 3x10^8 m/s in 350 days if you have sufficient thrust and power.

I'm pretty sure your calc is wrong. A confusing way to see this is a kind of Zeno like argument for relativity.

Warning: A long complicated overly complicated argument follows because I'm overly interesting in this and need to understand it.

Suppose I just accelerate to 0.5c. That would based on your calc take 175 days. Now lets consider you from the point of view of an observer travelling at 0.5c (O1)...he would think it would take another 175 days for you to get to 0.5c relative to him. Now consider a second observer (02) travelling at 0.5 c relative to 01...he would think it would take another 175 days to get to 0.5c relative to him. At this point what speed are you travelling at relative to earth rest frame?

Your speed relative to 02 is 0.5c.

Your speed relative to 01 is u = (0.5c+0.5c)/(1+0.5*0.5)= 0.8c

Your speed relative to rest frame is (0.5+0.

/(1+0.8*0.5) = 0.928c

We can continue this calc:

(0.928c+0.5)/(1+0.5*0.928)=0.976

0.991803279

0.997260274

0.999085923

0.999695215

0.999898395

0.99996613

0.99998871

0.999996237

0.999998746

0.999999582

0.999999861

0.999999954

0.999999985

0.999999995

So you got to 0.92c and it took 525 days. Now this calc is not completely correct because you are not in the frame of the observers and so you will experience less time that each of them for each of your 0.5c speedups. But in each calc you are only within 0.5c of each observer and in this case time is reduced by at most a factor of sqrt(1-0.5^2) = 0.866. So it would still take you at least 525 days * 0.86 = 454 days. And to get to 0.999657123 it would take you 175days * 8 * 0.866 = 1200 days.

But you are right it does take a lot less than 44 years to reach the required speed. Based on calc above you need 18 speedups to get to 0.999999995..which takes at least 175*18*0.866= 2727.9 days. At this speed the galaxy is only 10 light years across and will only take 10 years to traverse. So its doable.