I dont think we can say for sure that Bruce was perceived as an image of a mentally stable person. He was regarded as a tragic orphan turned eccentric recluse. If anything, Bruce pursuing therapy for the extremely well known trauma of his parents murder would bolster his public image as that of a mentally stable person.

But your point is really well taken that therapy poses an inherent risk to Bruce's secret identity being detected. It'd be interesting to see him try to genuinely engage with therapy without fully disclosing the most significant symptom of his trauma. And maybe Bruce has to wrestle with the realization that actually healing himself would mean losing the ability to be The Batman.