You mention dual-reporting to a functional head - if by "functional" you mean "process owner," then I applaud you for your intuitive foresight. In the field of process management, this is a common matrix management structure, and one that tends to work well, without diminishing a General Manager's autonomy.
GMs, as hierarchical managers, are interested in managing many aspects of an individual's career - hiring, development, compensation, appraising, and sometimes terminating. A matrix structure where an individual also maintains a process owner (PO) diminishes none of that capacity, while occasionally sharing appraisal responsibilities between GM and PO.
What makes this matrix arrangement worthwhile is the different mindset of the Process Owner - he/she is interested in the [external] customer, who provides revenue and profits for a viable organization. Focusing on the organization's various core processes ensures that the customer is well-served by the process workers, who collectively report to multiple GMs. Thus, the process worker is "served" by a GM who is interested in his/her career, and by a PO who emphasizes the importance of keeping the customer satisfied.
The additional benefit of this matrix arrangement is that it allows cleanly delineated process improvement efforts: when processes fail to satisfy customers consistently, the process owner appoints an ambitious and well-trained project manager to form a core team of 4-7 process workers to create a business case, gather data for root-cause analysis, and generate and implement creative solutions to reduce or eliminate the root cause(s) permanently. The advantage to the process owner is an improved process that he/she can then sustain through a strong process management mindset.
A final note: in some process-centered organizations, GMs and VPs also serve as core Process Owners, and so don't ignore the opportunity to make some of your senior leaders dual-hatted. You're on the right track!