Enabling Large Language Models (LLMs) to perform tool-free Table Question Answering (TableQA) presents two critical challenges: "question-first" data synthesis often introduces logical hallucinations, while standard reinforcement learning (RL) suffers from sparse rewards over long reasoning chains. We propose ProbeTQA, a unified framework that couples execution-verified data construction with process-supervised RL. To mitigate synthesis hallucinations, ProbeTQA adopts an answer-first executable probing paradigm, in which an agent performs multi-round exploratory programming over real tables so that every synthesized instance is grounded in a verifiable witness program. To address reward sparsity, we convert verified code trajectories into natural-language reasoning traces and derive execution-grounded process rubrics directly from runtime states, providing reliable dense supervision. Using these rubrics, we optimize the model with Group Relative Policy Optimization (GRPO) under a Two-Stage Reward Curriculum, thereby internalizing complex tool-use reasoning into the model parameters. Experiments show that our 8B model achieves state-of-the-art tool-free performance on challenging benchmarks, attaining 65.8% accuracy on TableBench and outperforming GPT-5.4 and DeepSeek-v3.2 by 13.4 and 6.8 percentage points, respectively.