The Exit function on the main menu of flwm_topside is hard coded into the binary, the 'System Tools' > 'Exit' is what brings up the Tinycore Exit menu like in Icewm. In Icewm, the Logout function is basically the same as the Exit on the main flwm_topside menu, coded in and brings about the same exit to the TTY. System Tools is the place for the expected Tinycore functions. Thanks for looking into this.
EDIT: I see that flwm_topside with Core has the main exit button bring up the Tinycore exit menu. I will fix this to have the expected behavior.