Future 6G heterogeneous wireless networks (HWNs) are expected to support multiple radio access technologies (RATs), dynamic wireless environments, and applications with diverse quality-of-service (QoS) requirements. In such environments, network selection (NS) cannot rely only on instantaneous radio measurements or static ranking rules. Instead, access decisions must account for the evolving wireless state, service intent, packet-level QoS behavior, and candidate-RAT dynamics. This paper proposes a large language model (LLM)-based digital twin (DT) framework for stable, application-aware RAT selection under candidate-set evolution. The main idea is to shift NS from an instantaneous decision-matrix operation to a decision process over an evolving wireless DT state. The constructed DT combines site-specific geometry, Sionna RT-based propagation descriptors, ns-3 packet-level QoS emulation, service context, candidate-RAT information, and decision memory. Rather than acting as a general-purpose controller for 6G networks, the LLM is used for DT-grounded decision intelligence in this specific NS task. On top of this DT, a unified intent agent translates user and service requirements into structured decision priorities for two complementary NS branches: an LLM-assisted multi-attribute decision-making branch (MADM--LLM--NS) and a direct LLM-based ranking branch (LLM--NS). To improve decision stability, the framework further introduces history-aware adaptive normalization (HAAN) and DT-memory-driven retrieval-augmented in-context learning (RA--ICL). Numerical results show that the proposed framework reduces rank-reversal problem and unnecessary handover events, while improving service-aware QoS satisfaction compared with representative MADM-based NS baselines.