For your point (1), one could incorporate into your algorithm a test to see if the numbers are unique and adapt your algorithm accordingly. I think that not only will they always be unique, but they seem to be a set of prime numbers plus 1 which I think has a bearing on possible approaches.
No need to make assumption 2, although I think it is in fact true. Testing that the remaining compartments once you've allocated the footwell need only progress to numCompartments-1. i.e. there is no need to test for the last compartment since if you've divided your weight totals by 3 and have allocated 2 of the 3 successfully, the final one must, by definition contain an equal weight. You can speed things up further in the testing compartment 2 because you don't have to find the actual set(s) of weights, but simply determine if there is at least one such set. An algorithm for that can be done in polynomial time rather than exponential with an appropriate approximation coefficient c:
https://en.wikipedia.org/wiki/Subset_sum_problem#Polynomial_time_approximate_algorithmI am no expert in number theory, but I suspect the fact that the weights are primes and one means you can take advantage of some kind of prime partition property that may prove that assumption 2 is in fact correct, but this is only a suspicion.