Purely looking at the chipsets, I'd go with the LX. It's limited to 66MHz (in-spec, and no great overclocker either) FSB, but it's a very efficient design with few stability or compatibility issues. Also, it can handle twice as much SDRAM as per official Intel specs - Intel states max 128MB per DIMM for 512MB total, but it can handle the same 16Mx8 chip 256MB DIMMs the i440BX can for a total of 1GB with 4 slots.
The ApolloPro was clock-for-clock much slower, with iffy drivers at the time. I believe you can use later 4-in-1 drivers to improve that a bit, but it remains slow,
As for the boards - only supporting Mendocino is pretty obvious when you have a PPGA socket and a chipset incapable of more than 66MHz FSB. That's not a board limitation but a platform limitation. That said, given you have Mendocino CPUs up past 500MHz, their performance was fairly competitive with their big P2/P3 brothers, and the LX is much more efficient than the ApolloPro, it's entirely possible that the fastest CPU you could stick on the LX board (assumedly a Celeron 533) would be in the same ballpark in terms of performance as whatever Katmai you could stick on the ApolloPro board.
Neither Zida nor Aska are exactly high-end brands. I'm aware of Zida's less than stellar reputation on their later Intel boards, but I'd hardly consider Aska a step up in any meaningful terms. Tbh, I'd just go by whatever CPU you want to run. If Mendocino, go ZIda, otherwise go Aska.