The 3100 and 3200 use Intel Atom C2000-series system-on-chip processors, which are subject to a hardware bug called AVR54. As data is transferred over the low-pin-count bus which connects the SoC to the boot ROM, the processor side of the bus link degrades. Eventually, the SoC can no longer get its boot firmware from the boot ROM, which renders the system unbootable. That may be what happened here.
Not sure which processor the 3100 has. Pretty sure the 3200 uses the C2558. 3600 is a C3558, and 3800 is a C3758. Interestingly, all the Atom C2000 and C3000 SoCs support ECC memory. Check Point's 1U boxes (4000, 5000, and 6000 series) mostly use consumer-grade processors from Intel (Core i family). Intel holds ECC-capable memory controllers back for their server-grade processors as a market segmentation strategy. With ECC memory, the 3600 and 3800 would be substantially more reliable than the bigger boxes with consumer-grade processors.
As for reprogramming the BIOS, if the TAC isn't able to help you out, it's definitely possible at a technical level. The boot ROMs are between the SATA connector and the drive power socket in a patch of the board with the silkscreen labels "BIOS 0" and "BIOS 1". They have small dots of paint on the pin indexes. With a set of SOIC jaws and an EEPROM programmer (such as a TL866II Plus), you should be able to dump their contents and write new contents to them. If the SoC has fallen victim to AVR54, though, that won't help.
If this is a box for business use, I would strongly recommend replacing it with a 3600 or 3800.
