For a new OSS project, I came across the AGPL 3.0 license ("GNU Affero General Public License 3.0") for the first time, and I'm a little unsettled now as to what license I should (and legally can) choose for my project.
Just for background, my project uses both the iText 7 PDF library (AGPL 3.0), as well as the Flying Saucer iText 5 renderer (itself LGPL 2.1), which internally uses iText 5 (again AGPL 3.0). My project also uses additional libraries licensed under ASL 2.0, EPL 2.0, and BSD 2-clause licenses.
From this question I gather that my project needs to be licensed under AGPL 3.0 because it uses AGPL-3.0-licensed code. However, that seems to be contradicted by the Flying Saucer iText 5 renderer using LGPL 2.1 while also utilizing AGPL-3.0-licensed libraries (unless their choice of license is incorrect/illegal, which I'm not really sure about what consequences that would have for my own project?!). Assuming that (from an AGPL 3.0 compatibility point of view), both AGPL 3.0 and LGPL 2.1 are an option, would either one raise incompatibilities with the ASL-, EPL-, and BSD-licensed libraries that I'm using?
In short, my question is: how do I determine what license options I have for a project that uses libraries licensed under AGPL 3.0, LGPL 2.1, ASL 2.0, EPL 2.0, and BSD 2-clause?
(I don't expect anyone to do my work for me, I'm happy to do the necessary research if someone could just point me towards how I would go about that in the first place)