Our little boy started getting a flat head as early as when he was 2 weeks old! We were told at the time that if we didn't work hard at trying to rectify it, a helmet may eventually be necessary.

I did a great deal of research and saw several specialists and nurses in an attempt to address the issue before it got "helmet worthy" and this is what they suggested (some things are not SIDS friendly, so use them or don't use them):

The following was suggested, but just didn't work for us:

If bubs isn't too mobile, look at getting a lovenest pillow.
Swap baby around in cot so that it encourages them to face opposite directions. This did not work - our boy didn't care about looking at the window!
Stacks of tummy-time. Again, didn't work for us as he hated it (and still does!)
Roll a towel up and place under bub's shoulder on the flat side. This will encourage them to face the other way. He rolled off it every time!

This is what actually worked for us. His head is now 100% normal:

Visit a paediatric physio and chiropractor. The most likely reason that bubs is favouring one side is due to having a stiff neck and/or weakened neck muscles due to their positioning in-utero. The physio will give you strengthening exercises to do with bubs and the chiro will adjust their neck ligaments accordingly.
Sleep bub on a lamb skin so that the surface under their head is softer than a firm mattress.
Sleep bubs on their tummy facing their head on the opposite side of the flat spot.

I know how very stressful this is and I really hope that you can avoid the expensive and stressful helmet route. Big hugs - I know exactly what you are going through