Accurate segmentation of tissues and instruments in surgical scenes is annotation-intensive due to irregular shapes, thin structures, specularities, and frequent occlusions. While SAM models support point, box, and mask prompts, points are often too sparse and boxes too coarse to localize such challenging targets. We present SCISSR, a scribble-promptable framework for interactive surgical scene segmentation. It introduces a lightweight Scribble Encoder that converts freehand scribbles into dense prompt embeddings compatible with the mask decoder, enabling iterative refinement for a target object by drawing corrective strokes on error regions. Because all added modules (the Scribble Encoder, Spatial Gated Fusion, and LoRA adapters) interact with the backbone only through its standard embedding interfaces, the framework is not tied to a single model: we build on SAM 2 in this work, yet the same components transfer to other prompt-driven segmentation architectures such as SAM 3 without structural modification. To preserve pre-trained capabilities, we train only these lightweight additions while keeping the remaining backbone frozen. Experiments on EndoVis 2018 demonstrate strong in-domain performance, while evaluation on the out-of-distribution CholecSeg8k further confirms robustness across surgical domains. SCISSR achieves 95.41% Dice on EndoVis 2018 with five interaction rounds and 96.30% Dice on CholecSeg8k with three interaction rounds, outperforming iterative point prompting on both benchmarks.