This paper investigates the finite horizon risk-sensitive portfolio optimization in a regime-switching credit market with physical and information-induced default contagion. It is assumed that the underlying regime-switching process has countable states and is unobservable. The stochastic control problem is formulated under partial observations of asset prices and sequential default events. By establishing a martingale representation theorem based on incomplete and phasing out filtration, we connect the control problem to a quadratic BSDE with jumps, in which the driver term is non-standard and carries the conditional filter as an infinite-dimensional parameter. By proposing some truncation techniques and proving a uniform a priori estimates, we obtain the existence of a solution to the BSDE using the convergence of solutions associated to some truncated BSDEs. The verification theorem can be concluded with the aid of our BSDE results, which in turn yields the uniqueness of the solution to the BSDE.