This paper investigates the joint dynamics of remittance corridors, foreign direct investment (FDI), and macroeconomic convergence between 38 Sub-Saharan African (SSA) economies and 27 European Union member states over the period 2000–2023. We propose an entirely novel estimator — the Bidirectional Heterogeneous Threshold Network Regression (BHTNR) — which, for the first time in the empirical literature, fuses three previously disjoint methodological traditions: (i) threshold regression with endogenous regime switching, (ii) multiplex network econometrics capturing bilateral financial linkages, and (iii) bidirectional panel error-correction modelling that permits asymmetric long-run adjustment between sender (Europe) and receiver (SSA) economies simultaneously. Unlike conventional gravity-augmented panel models or single-equation threshold VAR approaches, the BHTNR estimator identifies regime-dependent network centrality effects, maps non-linear transmission thresholds in remittance-to-FDI substitution, and recovers heterogeneous convergence speeds across dyadic country pairs. Estimated using a penalised iterative network least squares (PINLS) algorithm that we derive from first principles, our results reveal three key findings. First, remittances and FDI exhibit non-linear substitution dynamics governed by a remittance-intensity threshold of 4.7% of recipient GDP, below which they are complements and above which they crowd out FDI inflows. Second, network centrality in the SSA–Europe financial corridor amplifies economic convergence for hub economies (e.g., Nigeria–UK, Senegal–France) but generates periphery exclusion for low-network-degree SSA countries. Third, macroeconomic convergence is statistically significant only among SSA economies with institutional quality above a second estimated threshold, with speed of adjustment twice as fast in the upper regime. These findings carry profound implications for EU–Africa development finance architecture, post-Cotonou partnership frameworks, and the design of migration-investment complementarity policies.