"This will be my first BA job - which offer should I accept? "
Where do you have an opportunity to learn more ... not just BA stuff, but End-To-End Project activities ?
I do 100% Contracting, so my opinions might be different than a full-time person ... even though I do project management most of the time, I still do everything from BSA, Developer, Operations, Training, etc. Whatever they need at the moment to keep things on schedule.
Just don't get hung up on one thing either ... Agile, Waterfall, Scrum, Kanban, etc.
Many companies either have their own in-house version of something ... ranging from Terror to some type of well structured Hybrid methodology.
How stable is the Start-Up ? Do they have the clients to keep you employed ? Are they using Agile as a quick way to do ad-hoc projects ? Will you constantly be under pressure doing projects ? Who will be your co-located agile team ... and so on.
I would probably start with the Established Company to learn Concept to Implementation projects, documentation and working with various departments (technical, non-technical resources) and so on .... then when you get experienced with all that ... you might feel safer in a start-up.