If the 440BX satisfies the needs for that system, then I'd use it.
I wouldn't be very interested in the Apollo Pro 133. I can't remember if I've ever used the 133 though, so maybe it's okay.
If that's the Apollo Pro 133A (with the 694X chip part number), then I'd be more in favor of it. 133A has some advantages over Intel, but the plain 133 probably doesn't have as much going for it.
I don't have proof but my impression has been that VIA gradually improved with later chips. Thus far I've only had good experiences with the 133A, even though I love the BX I'm finding the 133A has some appeal.
I had a terrible time on Win98 with an old VIA MVP3 board (when it was current). This made me avoid VIA. But I've realized that the 133A boards I've used haven't given me any problems, at least with WinXP SP2. To some extent I think the chipsets improved, but also I think that over time, the problems with VIA had been identified and smoothed out in drivers. At this point we have more mature drivers than what was available back then. But again, my positive experiences were with the 133A, I don't remember using the 133.
440BX advantages:
-AGP on the BX is always trouble free, it might as well be considered the reference implementation.
People had problems with AGP on VIA, but I wonder how much newer drivers might have fixed this. I personally have had no trouble with AGP on the 133A, but that's with XP SP2 and nVidia 45.23, not Win9x. I don't know what things are like on the original 133 using newer drivers.
-PCI performance is better on the BX. Probably not a big deal for desktop use, but if it's doing a lot of I/O, stick with the BX.
-There might be USB problems on the VIA boards, I'm not sure, haven't tried that out much. 440BX USB definitely works. But if you want USB 2.0, you need a card either way.
There are some advantages to VIA P3 boards, but I think they mainly apply to 133A:
-Many support 512MB per DIMM - the 133A definitely does, the plain 133 might also but I'm not sure. Useless for Win9x/DOS.
-The 133A boards have Universal AGP slots, but the plain 133 and the BX are both 3.3v only. Anything that won't fit a 3.3v slot probably doesn't support Win9x anyway.
-133/133A both support 133MHz bus without any overclocking.
Many BX can also do this, and when doing so the BX is faster than anything. But how well this works depends on the motherboard. You need a board with a 1/4 PCI setting, and not all BX boards have that. A BX @ 133MHz will also overclock the AGP to 89MHz and this can't be fixed. Most nVidia cards of the time period don't seem to mind, but it's something you have to think about in the card selection. From the comments I've read online, it seems it pretty much always works on Geforce1-3, and usually works on Geforce4, but fails on FX5 and later cards. I was 3/3 using 89MHz AGP on a GF2MX, an early GF3, and a Ti4200. I don't know as much about ATI.
That all said, since this is for DOS and Win9x games, the features of the BX are probably plenty sufficient.
If you're targeting a higher specced build, VIA can be worth considering, but if not, use the BX.
You might also want to check which of those boards have options to disable the caches, if you want to slow it way down for DOS games.