Since we aren't part of the actual ICS contract, will we be able to actually contribute to the software sustainment mission? What areas of the FAR or USC specifically prohibit us from participating in the contractor's work?
Sounds like what you are thinking of is a Work Share agreement, type of public-private partnership that takes place at Centers of Industrial Technical Excellence (CITES), governed for the most part by USC 2474 (along with 2466, 2469, and 2208).
PPPs are structured primarily in one of two ways 1) Direct sales -- Govt pays the contractor, who quasi contracts with the depot and pays them (likely in WCF) or 2) Workshare, Govt pays the contractor via contract and also pays the depot via MIPR or other financial transfer between govt entities. In both cases, the organic depot does some of the depot maintenance
However, it sounds like there is a contract for equipment, training, etc. and another (ICS) for manpower; and no requirement for organic depot to do any of the maintenance activities, in which case it sounds like a lease agreement (which is still a PPP, industry using depot facilities, equipment, etc).
There is nothing that states that the organic depot has to have a piece of the maintenance activities, unless it is CORE workload (as per USC 2464), but the question doesn't say it is CORE.
The attachment provides a quick synopsis of a PPP, along with the PPP guidebook, https://www.dau.edu/tools/t/DoD-Public-Private-Partnering-(PPP)-for-Product-Support-Guidebook.
The last part of the question asks "what areas of the FAR or USC specifically prohibit us from participating in the contractor's work". It is not the FAR or the USC that prohibit it. The contract specifies what we need the contractor to do. If there is no PPP or other requirement levied on the depot, it is the lack of that requirement that will keep them from participating in the work.