Speech-based detection of cognitive impairment offers a scalable, non-invasive screening, yet algorithmic bias across demographic and linguistic subgroups remains critically underexplored. We present the first comprehensive fairness analysis framework for speech-based multi-class cognitive impairment detection, systematically evaluating bias mitigation across architectures, and demographic subgroups. We developed two transformer-based architectures, SpeechCARE-AGF and Whisper-LWF-LoRA, on the multilingual NIA PREPARE Challenge dataset. Unlike prior work that typically examines single mitigation techniques, we compared pre-processing, in-processing, and post-processing approaches, assessing fairness via Equality of Opportunity and Equalized Odds across gender, age, education, and language. Both models achieved strong performance (F1: SpeechCARE-AGF 70.87, Whisper-LWF-LoRA 71.46) but exhibited substantial fairness disparities. Adults >=80 showed lower sensitivity versus younger groups; Spanish speakers demonstrated reduced TPR versus English speakers. Mitigation effectiveness varied by architecture: oversampling improved SpeechCARE-AGF for older adults (80+ TPR: 46.19%=>49.97%) but minimally affected Whisper-LWF-LoRA. This study addresses a critical healthcare AI gap by demonstrating that architectural design fundamentally shapes bias patterns and mitigation effectiveness. Adaptive fusion mechanisms enable flexible responses to data interventions, while frequency reweighting offers robust improvements across architectures. Our findings establish that fairness interventions must be tailored to both model architecture and demographic characteristics, providing a systematic framework for developing equitable speech-based screening tools essential for reducing diagnostic disparities in cognitive healthcare.