What limits KV-cache compression at extreme bit-rates? We argue that it is not the choice of compression scheme, but how its budget is allocated across attention heads. Existing methods apply rank and bit-width uniformly, ignoring that each head has a different optimal mix of rank truncation and quantization. We show that co-optimizing rank and bit-width per head, using only standard low-rank projection and scalar quantization, dominates uniform allocation, with the largest gains at low bit-rates. Our method, KV-COBRA (Co-Optimized Bit-Rank Allocation), formalizes this as a resource-allocation problem: it balances rank-truncation loss against quantization loss within each head, then redistributes budget across heads to minimize total distortion. A fused Hadamard rotation equalizes per-channel variance, and reordering the SVD basis by attention-KL importance makes the solver query-aware. The same allocator extends to joint K+V compression. On perplexity, zero-shot, and long-context benchmarks from 0.5 to 4 bits per dimension (bpd), KV-COBRA shows the smallest accuracy degradation among evaluated methods at low bpd, with no per-token overhead.