In the end, the decision was made to move the Soldier to a safehouse.
Steve had protested throughout the entire discussion - what if something went wrong - but Natasha sided with Fury over the fact that several things already had gone wrong, and to continue like this was not something of benefit for anyone involved. Steve was compromised by the very presence of the Soldier, and every time Natasha opened her door to find Captain America with a fresh gash or other injury, she knew the time for his former best friend was quickly coming to an end.
Cuts and bruises were something the team could handle easily, and Natasha knew that Steve had sustained more than his fair share from the Soldier. Those things weren't exactly beneficial to morale, but they were manageable and would fade over time.
HYDRA and their weapons, however, were another story. It was a thorn in her side, how slowly the remaining members of SHIELD were moving to any sort of action. Decisions were fine, but after they were made, Natasha's perspective was that the actual doing something should follow almost immediately after.
The agent that had attempted to eliminate both Steve and the Soldier had been a clear warning to the team; HYDRA knew the Soldier's location and would eventually come to collect what they felt was theirs.
The level of conditioning that had been ingrained into the Soldier's mind wasn't improving the situation, either; any words that sounded vaguely like an order would cause him to snap back from any progress and become a blank slate again. Even mundane things like the pop of the toaster or the faint hum and buzz coming from the lab would have the Soldier flinch and revert to his old ways.
Steve and Natasha were voted as escorts to bring him to the safehouse and stay there with him until the immediate threat had passed. The morning of their departure was the toughest one yet; neither of them had been able to sleep and the situation was becoming more and more tense. The Soldier's screaming had increased to a new level during the night and Natasha had honestly considered shooting through the floor so she could have some peace.
Against Steve's wishes, Natasha did the same thing to the Soldier that she had done to Clint; just a bit of cognitive recalibration to keep him subdued during the trip. He'd refused any needles and had become a wild thing whenever the subject came up.
As much as she knew that splitting the group apart was a risky thing to do, Natasha hoped that - for Steve's sake, at least - there would be some benefits to outweigh the dangers.