To be fair, some dedicated chess computers were made with very limited hardware - my family once had a
Tandy 1450, which apparently was literally built around a 4-bit
elevator controller-type microcontroller. The 1450 refers to the manufacturer's claimed ELO rating for the machine, the actual rating measured by subsequent collectors being substantially lower, and 1450 being decidedly unimpressive in the first place. But it was running in 4KB ROM, less than a kilo-nybble of RAM and with a 600kHz clock…
On the Gigatron, I suspect that code size will be less of a problem than execution speed. This means you can compensate for the inability to search deeply (even 4 plies / 2 moves will be a struggle at this level) by implementing a relatively sophisticated position evaluator. First check for mates or situations that make the move illegal, then add up material scores, add a factor for mobility (the number of pseudo-legal moves that can be made from that position), penalise doubled and/or isolated pawns, and penalise pieces that are attacked without being supported. Subtract Black's score from White's, implement a basic alpha-beta negamax search, and you have a basis for making not-completely-stupid moves at even 2-ply search depth, though 4-ply will add the ability to evade the simplest traps (at the expense of requiring more thinking time).
Interestingly, it seems that some of the strongest chess computers of the 1980s were built around a 6502 - sometimes with a large GAL whose function is unclear to me, but could include chess-specific acceleration functions. Today, the preferred platform seems to be a Cortex-M4 or M7 based microcontroller.