Since morning tea is not a regular meal where I live, I am having trouble visualizing how long this gathering is supposed to last and how much there is an expectation of everyone eating. Business where I live is more usually conducted over lunch, and one person sitting and not having lunch while everyone else is, would be very noticeable and seem odd.
If coffee and pastries were provided during a morning or afternoon break, with everyone just mingling about, nobody would even notice who was eating and who was not.
Are these long meetings? Do they consume your entire break time? Do you have to eat your regular morning meal at that exact time to the minute (because of break rules or something?)
Adapting my circumstances to my (limited) understanding of the situation, I would eat my own food at my desk a few minutes before the meeting, and just carry my drink of choice to the meeting so I could interact. Maybe not every day, but regularly enough to keep my face out there.