hi,
some news... i still can't tell what happen exactly, but it might be the TI 486 SXLC2 having some troubles with OS/2.
but i can't easily swap with the original 386, as i soldered the TI and don't plan roll back anyway.
Should not be a problem with the CPU on its own, I ran OS/2 v3 on an ISA board with a Cx486SLC for years, up to fixpack 4x, the only difference between that chip and yours is a bigger cache and optional clock doubling. If you can get hold of TI's utility software package that they distributed for the chip, you can try toggling the clock doubling feature on and off, but I doubt that would make a difference.
I sold a number of these as upgrade chips (on a socketed PCB typically) and as budget replacement motherboards in its day and while we did provide the TI software package to people, no-one ever seemed to use it since they were more or less plug and play.
IBM did supply updated firmware and configuration software for some of those ISA based PS systems, such as the PS/1 386sx, as they had some problems with booting OS/2 32bit but I have no idea if that applies to your system.