Arguably, cycle-accurate emulation should be done with FPGA support anyway. The Pentium III is old enough that there should be no legal obstacle to implementing new compatible hardware.
The problem won't be legal, but scale: the Pentium 3 has ~9.5M transistors [1] - and the world's largest FPGA, at a staggering price point of 55.000 dollars, has only 9M logic gates [2]. Probably you could use an FPGA a bit smaller if you use FPGA-native memory for L1-L3 instead of transistors, but you'll nevertheless need an FPGA that is many orders of magnitude too expensive for a hobbyist niche project. Also, you…
Far from a Pentium MMX.