Okay, let's be real: pets are not "family members" the way kids are. Pets are, first and foremost, a luxury item. In times of belt-tightening, it's true- the luxury expenditures can get cut.
That said, $140 isn't really all that much. $35 a week is pretty easy for most households to scrounge up or pinch off from their bills. If you're really bare bones everywhere else and can't find that much, then yes, the basic numbers crunch says you can't afford the dog. If the dog truly does need specialty food, well, there's your answer.
That's the practical/logical/"just the facts" side of it. As for the emotional side, your kids are going to lose the dog eventually anyway. Would it be sad to have to rehome it? Sure. But the point is, the kids will eventually go through the sadness/grief of losing the animal anyway. That's one of the dark gifts of pet ownership: lessons in loss. If you've made the decision to rehome, be direct with your kids, only giving them age-appropriate details (your 4-year old doesn't need to try to process "uncertain economic times" and may think they're next on your budgetary chopping block!). Be sure they know they're not going anywhere, and that the dog will be well cared for by its new family.
For the logistics, if you're going to do this, try to work with a shelter/rescue organization while serving as the dog's foster, rather than just dumping the dog at the shelter. It's MUCH easier to rehome a dog who is operating normally in a home environment. Many dogs deteriorate rapidly in a shelter environment, and poor kennel presence is a death sentence. Do right by your pet and agree to foster for at least one month, giving the shelter/rescue time to find the best placement for the dog. Alternatively, if you bought from a breeder, many breeders will take "their" dogs back, so that may be an option.
Ignore the idiots on this thread telling you you're a bad person for rehoming the dog. Not everyone's life goes as perfectly according-to-plan as theirs must always go.

Things change. If the pet is no longer a good fit for your family, it's not fair to the pet to try to force what isn't working. Rehoming correctly is a fine plan.