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.
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 would need serious quality photographs of the die... which can be done for old and small chips on a budget, but will be really expensive even for something of the Pentium 3 era.
What has always interested me though: how are CPUs and modern GPUs actually developed, given that FPGAs are way, way too small and silicon-making runs extremely expensive?
[1] https://www.computerhistory.org/revolution/story/330
[2] https://hackaday.com/2019/09/24/who-could-possibly-need-an-f...
Far from a Pentium MMX.
FPGA LUTs are quite sophisticated, the LUTs in both Xilinx and Intel's latest offerings are able to implement an arbitrary boolean logic function with up to 6 inputs and 1 output. The FPGA also comes packed with other specialized hardware such as on chip memory and multipliers so you do not need to burn logic to use those things. You very likely could fit a Pentium III on a moderately large FPGA.
The real challenge would be matching the clock speed of the original processor. Even 400MHz of the first pentium IIIs on the market might be difficult, getting near 1GHz is likely impossible.
At least some of it is done on FPGA clusters, boards with multiple enormous FPGAs. Cost of such a system is in at least 6-7 digit range.
I wonder though about truly ancient x86, why aren't there more clean-room clones of those?