Don't really want to get into the taking things from breakfast debate, but walking out while still munching on a croissant or piece of fruit is fine- making up sandwiches and filling your lunch box is a bit
IMO.
OP are the half board vouchers you have Premium level? these are usually the ones offered to Disneyland hotel guests and will fully cover any meal you have at any of the Disney restaurants (except brunch in Inventions on Sunday only) with only the cost of drinks to pay on top. however as you will have paid extra to have Premuim vouchers it would make sense to only use them at the most expensive restaurants, otherwise you will effectively be paying €60 odd for a meal from a €35 menu.
These restaurants are Inventions at DL hotel- already mentioned here, fab meal, extensive buffet and characters visit your table; Auberge de Cendrillion in Fantasyland-this is the meal with the Princesses, traditional French cuisine; Buffalo Bill's wild west show in Disney Village- dinner is served while you watch this spectacular show. California Grill, also at Disneyland hotel, would be covered by your vouchers too but this is a proper posh restaurant (only place at DLP with a dress code) so possibly wouldn't be suitable for you little ones, tbh I'd just do Inventions again instead!
As well as your meal vouchers you will also get a ticket that can be exchanged for a tea time treat (3-7pm) at many places in the parks. I find with a big breakfast and dinner, all we need are a few snacks in between and never stop for lunch. At the most I would maybe grab a sandwich or take my own when self-catering, but you could choose to get a quick counter service meal which usually cost around €10-15 per person for a burger/cips/drink type meal. Depending on when you are going in September, not everywhere will be open for this though as it is low season.
Finally i would make reservations before you go for any places that you definitely want to eat at, to be guaranteed a table and avoid long waits. Even during low season restaurants get busy (especially the character meals) as a lot of people want to eat at the same time, particularly after the parade and before the Dreams show in the evening. I'm going in September too and have already booked all my meals.
P.s €2000 is plenty to take- that is more than my whole trip is costing me!