We stay at Travel Lodge in Ashton - about 10 mins away from Eurotunnel. They do serve breakfast there but possibly not that early.
To be honest if its so early will you really want breakfast. We often take cereal bars or hot cross buns and a small tub of butter and either eat these on the train or in the car at the other end. You can also get breakfast at the terminal - not sure what time things open but we got a croissant there at 6.00ish and they were serving hot breakfasts as well.
The other option is just to get to France then stop somewhere on route for breakfast- most of the service stations do a breakfast of drink, croissant, roll and jam