Your problem is with Office Authentication and FBA. Either turn off the
Office Client Features (Authentication section) or have users check the Sign
Me In Automatically box. We ran into the exact problem. The issue  that with
FBA, credentials are not carried across application boundaries unless you
use cookies and have the cookie persist (check the box). Its a bit of a pain
but that's the only way that I know of.

I'm pretty sure you are running into the standard problem of FBA not
supporting Client Integration. It happens because the FBA user  doesn't
have a Windows Identity on the SharePoint server and therefore can't use
the SharePoint OpenDocuments control to start an editing session on the
client. There is no real fix  for it. There are various ways you can
have the FBA user linked to a Windows Identity so that you can turn
client integration back on. But most people simply solve it by training
people to use the context menu  and select Open.

Since we were working with roaming profiles we had no problem after the user
checked the sign me in box. We set the FBA session to expire after 24 hours
(I suppose you set it to never expire) so our user  only logged in once a day
if at all. This allowed Word and the Office apps to use the credentials in
the cookie and authenticate the user giving full office integration with
FBA. We sort of stumbled on this fix.

Clicking the remember me box creates a permanent cookie that ties the
user to the Computer Account on that workstation. The computer account
is a legitimate Windows Identity so Client Integration works.