In my experience stick with your first choice.
You made that choice for the right reasons at the time; some were objective, others instinctive.
When you go through a tough time – which all jobs have – you'll think that the original job would have been better.
Go with your gut feeling.
What you are describing is the standard life of a BA, so maybe you have selected the wrong line of business if you dislike it that much. Sure, in your case there are more issues due to unclear business requirements, but that is to be addressed by the PO (Product Owner) that is not doing their job as expected. One more thing: the BA is not a QA Engineer, so ...
Tell the backup company you need ten days to take care of personal business before you can give them an answer.
It depends on the industry and level, but for professional jobs this is normal enough that if they cannot accommodate you then you have probably dodged a bullet.