Language-model watermarks can help identify model-generated text, encode provenance information, and detect post-generation edits. A broad class of logit-based watermarks increases the logits of a designated token set at each decoding step. Because a base model's native probability mass on this set varies across contexts, models, and watermark rules, the same logit bias can yield different adherence rates and increases in perplexity (PPL). We introduce watermarkability to characterize the compatibility between a model and the token sets requested by a watermark. At a target mass $τ$, we quantify this property through a watermarkability cost: the expected minimum KL divergence from the base-model distribution required to assign probability at least $τ$ to the requested set. A lower cost means that the model is more watermarkable for that request distribution. To reduce this cost, we propose a two-stage method. In the first stage, we train a target-set-conditioned low-rank adapter on unions drawn from a 16-way vocabulary partition to increase the requested set's probability mass. In the second stage, a closed-form controller applies the smallest remaining uniform logit bias needed to reach the target mass. We evaluate the method on six watermark schedules using OPT-125M and Qwen3-8B. Across the 12 model-schedule pairs, the pre-adaptation watermarkability cost correlates with the log-PPL increase under a near-hard fixed-bias baseline (pooled Pearson $r=0.929$). For every schedule, at least 99.80% of sampled tokens fall in the target set requested at their decoding step. At this reliability, the adapter and controller reduce average per-schedule prompt-conditional PPL by 46.3% on OPT-125M and 5.2% on Qwen3-8B relative to the fixed-bias baseline.

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