Use Google Tasks in Thunderbird

If your mobile phone is supported, you could run Funambol mobile client on your phone and install the Mozilla sync plugin into Thunderbird. Both sync the local calendars and contacts with a Funambol server (you can use their service or run your own).

I use a Google Calendar with Thunderbird/Lightning and I have always been able to Accept invitations through Thunderbird.

I recently lost the Accept/Decline buttons and just figured out the problem. I had recently added a shared iCloud calendar, and the new calendar was automatically associated with the same email address as the Google calendar.

All I had to do is set the iCloud calendar to not be associated with my email address, and my Accept/Decline buttons came back.