Currently there are 6 backpacks and enough rope to secure 8 other containers. Maximising the animals to carry 12 total would leave two backpacks to be worn by dwarves, while everyone fills their belts and carries a container in their hands. That's basically treating them as slaves though, so a good rest would be earned.
4 Pack animals: 12 Medium
2 Backpack Dwarves: 4 Medium, 8 Small
Each Extra Dwarf: 1 Medium 4 Small each
If all possible space was used for bringing back food or drink on the way back, the animals and backpacks could carry 280 rations and hands and belts carry another 36 from each dwarf.
Building a new cart would allow an animal to carry 120 instead of 60, and free up another three backpacks for dwarves to carry.
4 Pack Animals, Cart: 15 Medium
5 Backpack Dwarves: 10 Medium, 20 Small
Each Extra Dwarf: 1 Medium 4 Small each
If all possible space was used for bringing back food or drink on the way back, the animals and backpacks could carry 400 rations and hands and belts carry another 36 from each dwarf.
Bringing Sandow's Golem would let it carry another three containers with ease.
Building a second cart would exhaust your forged iron supplies, but increase capacity by an additional 3 containers. With four pack animals and two carts, a golem, and six dwarves with backpacks, you could transport a grand total of 756 meals. In a minimum two-week travel period the 14 Dwarves would consume roughly the same total amount of food and drink, though replacing alcohol with water and foraging would reduce that.
Ultimately, the expedition would extend the food supply by two weeks, while forgoing all other production.
Your comically depressing situation would probably entice the Elves to help you transport more supplies, though.