You can try, but I personally think it'll be hard. Unless you're entering as MX11, else there is no way they'll offer you 6k+ (the salary ceiling for MX12 is around 5.8k).

To be honest, AH is just one level up from SO (entry level), and typically a fresh grad takes about 3 years to be promoted to AH level. The fact that they are offering you AH position could mean that they view your experiences as insufficient to be considered for middle management role (Hd - DD level). Then again it's more important to look at the grade which they are offering you (EDB may have different grade scheme as Ministry).

I personally have more than 20% increment when I jumped from a stat board to a Ministry. Though the reasons I could do that was because it is hard to find candidates with relevant experience for my role, and I could "negotiate" as I have offers pending my reply from other agencies.

I would say no harm in asking, they probably have a limit as to how much they can offer you (all HR people would know this). If your request is too much, they will counteroffer. Last word of advice, counter their counteroffer with an additional $100 increment (that's usually their last last buffer).