The PCjr ROM will not work in an IBM 5150/5160. The IBM 5150 and 5160 ROMs do a check of the timer and IRQ early in the POST which will result in the BIOS just halting if it fails. This code executes before anything is displayed to the CRT, so maybe this is where you are getting stuck on these ROMs...Also as a curiosity, I have tried other ROMs, such as the original IBM ROMs like ibmjr, ibm5150/5160 and they don't work.
My suggestion would be to use the IBM BIOS versions and sticking with it until you can boot to DOS. I think this is a good place to fail because the BIOS is published in the technical reference and you can trace the code where the POST is getting stuck which is much easier than when games get stuck as they are nearly impossible to trace at that point. This is how I did initial debug on the MCL86.. with multiple versions of IBM BIOS, and even the fixed disk controller's BIOS!
The MCL86 core and BIU are fairly well debugged and work with every version of IBM and third party BIOSes I have tried, so I suspect there is an issue with the emulated peripherals like your 8288 and he 8259. The real 8288 (and my BIU) use the rising edge of the CLK at the beginning of the cycle and negative edge in other places, so if your core does not support this the bus cycle could be corrupted. For interrupt cycles, the BIU will perform a double-read, so maybe your 8259 core is not supporting this correctly, or is providing the incorrect vector to the CPU. Or maybe the interrupt priority scheme is not emulated correctly.. Just a few thoughts on where to look...
The MiSTer FPGA is a big/fast device, but is there a chance your FPGA timing is not constrained enough and you are getting timing violations?