With explosively escalating service demands, beyond fifth generation (B5G) aims to realize various requirements for multi-service networks, i.e., higher performance of mixed enhanced mobile broadband (eMBB) and ultra-reliable low-latency communication (URLLC) services than 5G. To flexibly serve diverse traffic, various functional split options (FSOs) are specified by 5G protocols enabling different network functions. In order to improve signal qualities for edge users, we consider FSO-based coordinated multi-point (CoMP) transmission as a prominent technique capable of supporting high traffic demands. However, due to conventional confined hardware processing capability, a processor sharing (PS) model is introduced to deal with high latency for multi-service FSO-based networks. Therefore, it becomes essential to assign CoMP-enhanced functional split modes under PS model. A more tractable FSO-based network in terms of ergodic rate and reliability is derived by stochastic geometry approach. Moreover, we have proposed CoMP-enhanced functional split mode allocation (CFSMA) scheme to adaptively assign FSOs to provide enhanced mixed throughput and latency-aware services. The simulation results have validated analytical derivation and demonstrated that the proposed CFSMA scheme optimizes system spectrum efficiency while guaranteeing stringent latency requirement. The proposed CFSMA scheme with the designed PS FFS-CoMP system outperforms the benchmarks of conventional FCFS scheduling, non-FSO network, fixed FSOs, and limited available FSO selections in open literature.