Hi Rob,
Not sure if you have done this, but you mention everything else aligned
except checking if you are ACTUALLY pointed at the SCP.
I wrote a guide on this and how to recognise the stars of sigma octanis in the articles section which could help you understand and recognise if when you have the scope "parked" at the south position you are actually close to the SCP.
Here it is. You will just need to spin everything 180 degrees from the pics as I did this when everything was up the other way and Achernar was in the south west, not the south east like it is now. But the method still works.
One reason I wrote it is because I checked and checked over and over but when I pointed my scope correctly at the SCP, I found my tripod was not pointed exactly at it AND, worse still, my latitude scale was WOEFULLY incorrect by about 15 degrees!
Take a look and see if you can recognise the SCP through the finder or even the main scope with the widest eyepiece you have, once you have the scope right, then look at the other things.
When you mention you point the scope to the bright star and then "sync" the program, I assume this is just about the same thing as when i do my 1 star alignment on my HEQ5 (or use EQMOD with Stellarium).
What I have noticed is that when I do that (and my scope IS dead on the SCP) through the polar scope and the finder etc) is that the goto on my mount is not as accurately synced as it can be and the results I get sound A LOT like yours, in the general area, but significantly off.
I use the 3 star alignment as a preference as the results are 10 times better when I do, something which I think you might not be able to do with what you have.
So therefore, see if you can determine that the mount is properly aligned and pointed at the SCP, THEN try your Iphone app and see if it improves. With only one star to align on, there is nothing to fine tune the system compared to the accuracy of using 2 or even 3 stars to lock you in.
Good luck with it, sounds like a cool setup, hope you get it sorted.
Cheers
Chris