You ask only about using the software; that is, you don't intend to modify it. Your primary concern is cost. That said, BSD, Apache, and MIT licensed software should be fine for you and your end-user to use without payment or any other (significant) obligation.
GPL2/3 are both trickier, because you are not the intended end-user of the software. When you convey the software to the end user, you will incur the obligation to provide them with sources. They, provided they do not convey it outside their organisation, will incur no further obligation. No one should incur cost, save the costs associated with providing sources.
AGPL is trickier still. Although no payment is required in order to use it, the source obligation of GPL applies to the conveyance from you to the organisation. Furthermore, if your use includes a web service which can interact with users, the operator of the service will incur the obligation to create a mechanism by which users can get copies of the source.
Numpy I am not going to consider. It is now well-understood in the community that crayon licenses are a bad idea, not least because they require much more detailed study to properly understand.
Edit: as Bart points out below, and I am grateful to him for so doing, the Numpy licence is 3-clause BSD with the serial numbers filed off, as it were. So you should be fine on that score, too. But the Numpy developers still deserve a slap for not simply using 3-clause BSD - licence proliferation is still a bad thing, even if they're all the same licence with lots of different names.
As ever, IANAL/IANYL. Philip Kendall is right that lawyers should be consulted if there is doubt; you might not have any, but the public institution should, and it needs to properly understand its obligations in order that it may fulfil them.
