A few thoughts come to mind:
(1) You mention being at the resort a lot for naps/resting. If it were me (and I have been in that situation in the past when my kids were younger
), I'd look at a resort with easy transportation with a stroller and young kids. Since you've stayed at all of the deluxes, you probably already have a good idea which resorts will fit the bill.
(2) Having said that, Poly would be my choice for your situation. You can easily get to MK and Epcot simply by rolling your stroller on and off the monorail. Easy to get back and forth between those 2 parks and Poly. Staying in Hawaii (CL) gives you an amazing view of Wishes. If you can swing it, book a lagoon or theme park view and while your kids nap in the room, you can sit on the balcony/patio with a drink while having one of the best views in WDW.
(3) As far as food for kids, the 4 year old would probably like the turkey corn dogs out most nights, PB&J, lots of fruit (pineapple, berries, melons), cheese, crackers, veggies. Not sure what the 1 year old will be eating at the time of your trip.
(4) Poly lounge stays open all the time, even when it's a non-food time.
(5) As far as adult food, I posted pictures of each night's offerings in the CL thread from our trip a couple weeks ago.
(6) Breakfast is pretty standard in all the lounges and you wouldn't have any problems finding food you guys like (cereal, bagels, muffins, Danish, some/most have oatmeal, yogurt, juices.....some have hard boiled eggs and meats (Poly does not).
(7) Lunch time has small snacks like goldfish, gummies, etc. Nothing major, just snacks.