Large Language Models (LLMs) are promising tools for synthetic data generation in mental health. However, privacy policies and restrictions forced previous work to rely mainly on generic information. We present a comprehensive corpus of synthetic therapist-client conversations generated through LLMs. We construct our generation pipeline, SQPsych (Structured Questionnaire-based Psychotherapy), which uses real structured client profiles and psychological questionnaires without leaking any sensitive data. We fine-tune various openweight LLMs on our generated corpus, SQPsychConv, and test them through both automatic benchmarks and human evaluation with trained psychotherapists. We find that standard benchmarks do not adequately capture the strengths of our dataset, but expert judgment shows that SQPsych makes LLMs significantly better at therapist roleplaying. Experts also consistently prefer therapy sessions generated by our models compared to other mental-health-oriented LLMs. We release our code, fine-tuned models SQPsychLLM, and corpora at https://ai-mh.github.io/SQPsych.html.