From your list the only thing I have used is Sage. Pretty good software. No direct connection that I recall. You actually need to import FM data. Currently I use a custom accounting solution in FM.
I Think quick books used to have trouble with "assemblies". I actually worked with a manufacturing company that left quick books because it would not allow them to have products made up of other products. Not sure if that is the case now.
We have used AppleScript to exchange data with AccountEdge and do things such as creating new AccountEdge contacts and generating AccountEdge invoices. Its not "integrated" per-se in that you can't create an external file reference to it but you can exchange data via a client machine as long as both FileMaker Pro and AccountEdge (or AccountEdge NE) are running.
I don't have an answer but I am interested in this also.
Having moved over to Mac I'm also in the hunt for core financials for my Business
Sales Ledger, Purchase Ledger, Cash Book and Nominal Ledger
Some others I have come across are
KiBiz FileMaker Accounting
But to be honest the only one that appeals to me at this stage is Moneyworks Datacenter.
Either by using the filemaker 32/64bit plugin as single user or REST API on the DataCenter version I prefer to have my accounts "local" to my business and not hosted in the Cloud.
New Millennium gets good recommendations here also, but I believe it can be expensive.
We're big fans on Xero and have used it for many years. We've actually built a solution that shows you how to integrate a FileMaker database with Xero:
We've got customers in the US, UK, NZ and Australia using this happily for several years now.
Like bigtom, I have developed a custom FM solution, so that is an option you might consider. I can't comment on the other options you've listed, except to say that I have heard someone sing the praises of Xero, but I don't know what they were comparing it to.
Thanks everyone! These are helpful comments.