Is they a way to make it produce a boot log?
Not really, besides installing the debug kernel and monitoring the serial port.
Since you got a reaction from running winos2.exe, it must have partially installed.
You can review your config.sys for lines like,
DEVICE=W:\OS2\MDOS\VXMS.SYS /UMB
DEVICE=W:\OS2\MDOS\VDPMI.SYS
DEVICE=W:\OS2\MDOS\VDPX.SYS
DEVICE=W:\OS2\MDOS\VWIN.SYS
DEVICE=W:\OS2\MDOS\VW32S.SYS
DEVICE=W:\OS2\MDOS\VMOUSE.SYS
DEVICE=W:\OS2\BOOT\POINTDD.SYS
DEVICE=W:\OS2\MDOS\VMANWIN.SYS
Though I can't remember which are needed by Winos2. Some of the device drivers might be in your Dos 6.22 directory I guess.
Also review your win.ini and especially system.ini. Unluckily it has been so long since I used Warp V3 Redbox, I'm not sure what should be in there and also how it works when you run it from OS/2.
My boot section in system.ini (Warp v4.52 bluebox but not much changed when it came to Winos2)
[boot]
useos2shield=1
os2shield=winsheld.exe
shell=progman.exe
mouse.drv=mouse.drv
network.drv=
language.dll=langeng.dll
comm.drv=comm.drv
keyboard.drv=keyboard.drv
system.drv=atmsys.drv
;386grabber=vga.3gr
;286grabber=vgacolor.2gr
drivers=mmsystem.dll
WAVDMAPPS=
atm.system.drv=system.drv
display.drv=ifgdi2vm.drv
sdisplay.drv=isgdi2vm.drv
fonts.fon=vgasys.fon
fixedfon.fon=vgafix.fon
oemfonts.fon=vgaoem.fon
os2user=user.exe
os2gdi.exe=GDI.EXE
os2user.exe=user.exe
It's possible that running winos2 uses a different system.ini too.