This paper investigates joint device identifica-tion, channel estimation, and symbol detection for coopera-tive multi-satellite-enhanced random access, where orthogonaltime-frequency space modulation with the large antenna array isutilized to combat the dynamics of the terrestrial-satellite links(TSLs). We introduce the generalized complex exponential basisexpansion model to parameterize TSLs, thereby reducing thepilot overhead. By exploiting the block sparsity of the TSLs inthe angular domain, a message passing algorithm is designedfor initial channel estimation. Subsequently, we examine twocooperative modes to leverage the spatial diversity within satelliteconstellations: the centralized mode, where computations areperformed at a high-power central server, and the distributedmode, where computations are offloaded to edge satellites withminimal signaling overhead. Specifically, in the centralized mode,device identification is achieved by aggregating backhaul infor-mation from edge satellites, and channel estimation and symboldetection are jointly enhanced through a structured approximateexpectation propagation (AEP) algorithm. In the distributedmode, edge satellites share channel information and exchangesoft information about data symbols, leading to a distributedversion of AEP. The introduced basis expansion model forTSLs enables the efficient implementation of both centralizedand distributed algorithms via fast Fourier transform. Simu-lation results demonstrate that proposed schemes significantlyoutperform conventional algorithms in terms of the activityerror rate, the normalized mean squared error, and the symbolerror rate. Notably, the distributed mode achieves performancecomparable to the centralized mode with only two exchanges ofsoft information about data symbols within the constellation.