This is a tough one...tough since there is not an easy answer.
Had they left within 6 months, it would be easy. Under the Uniform Child Custody
Jurisdiction And Enforcement Act you could file in LA and keep the case in LA.
After 6 months are you stuck in GA.
And you ask a great question...do you wait to show mom is unfairly denying you access? Or file sooner.
I would file sooner.
Here is why:
The court will decide this based on one thing, and one thing only. What is best for the child.
NOT what is best for mom or dad...the court cares little about that (when it comes to the custody order at least). The longer you wait, the more firmly the child is rooted in their circumstance, the more difficult it will be to get the court to change it.
I agree, you have a point that mom is not being fair to you. But at the end of the day the court is concerned not about you or the mom...the court is concerned about the child
That leads to your next point...the mom leaving the child to be cared for by grandmom...if there is evidence that this is not good for the child? You bet the court will consider that. And that could be the evidence you need to secure custody. Specifically, if you can show that mom is placing child with grandmother and that the environment is not good for the child, that is something the court will be very concerned about.
So you may want to file this case now (while this issue with mom leaving the state regularly) to press the issue. It may be you can get the court to give you custody based on that alone.