It looks like that your router is a non-digi product. If it is so, then on both coordinator and router you need to assign ‘ZigBee stack profile’ parameter ZS = 2. In case they both are digi products then put ZS=0 on both of them.
The scan channel should be the same on both devices, SC=0x7FFF is OK in the XBee (all channels); you should check that CC2531 has the same channels enabled.
Also, I have had some problems with XBee routers “creating” their own network on another channel when the coordinator is not present. I solved it by setting JV=1 (see this command’s documentation below from the XBee PRO datasheet).
As other users stated above, it is a better idea to use the same unique PANID on both devices instead of 0x00. If you want to use 0x00 check that OP (Operating PAN ID) and CH are equal on both to ensure that both devices are on the same PANID.
Finally, if the router does not connect, the AI parameter may help you determine what is happening.
JV Parameter: Channel Verification. Set/Read the channel verification parameter. If JV=1, a router
will verify the coordinator is on its operating channel when joining or coming up from a
power cycle. If a coordinator is not detected, the router will leave its current channel and
attempt to join a new PAN. If JV=0, the router will continue operating on its current
channel even if a coordinator is not detected
i also have a battery pack running through a 3.3V regulator (powering MCU and Xbee… When i attempt to run off the battery power after disconnecting the debugger the blinking led/ pwn work but the xbee no longer works… I have checked power / ground / DIn(from UART TX) and Dout(from UART RX) power/ ground good, transmit pin on xbee is getting signal but nothing is sent to x-ctu (which works perfectly fine while connected to debugger)