I don't have a 4 year old (my oldest is 2 1/2) so I'll try and give you advice based on my experience. Most kids don't really like sandwich bread to begin with - so would she be interested in other options such as pitas, flat breads or tortillas. (We had tacos for dinner last night and I made one for my toddler knowing she wouldn't eat it but trying to introduce it nonetheless. She wasn't interested in the 'taco' form, but when I unrolled it I got her to eat the tortilla because she had the liberty to 'tear' it which was fun for her.)
If your DD will eat salami, could you make her a sandwich on one of the options above - or just include some salami bites, along with some cheese chunks and some crackers (perhaps triscuits or wheat thins which are at least higher in fiber)? If she likes turkey or chicken - you could do the same and just get chunks of turkey/chicken from the deli instead of slices and cut up some bite size pieces for her (or make your own at home). I'm sure you've seen that they also sell chunked chicken/turkey in the prepackaged section of the store. The point is that the bread itself isn't important if you could get her to get her fiber in another form.
My DD also enjoys some raw veggies such as cucumbers (which I deseed and cut into sticks), carrots sticks (again I cut small sticks as the baby carrots are too thick and dense), and grape tomatoes. She'll eat these fine with some ranch dressing to dip in!
What about fruit - will she eat fresh grapes, bananas and strawberries? Yogurt?
If she can have hot stuff such as ravioli and soup - what about chicken nuggets, mac & cheese, turkey meatballs, etc.? I'm guessing she won't eat those though since you didn't mention it. What about the microwave grilled cheese or PB&J sandwiches?
Anyway - just some thoughts. Hope this helps.