Various approaches can do the job and, as noted, there should be no problem with a quad-split.
You might start with the main track and stack one other above it. Sync and mix them, then mute the 2nd track.
Add track three, sync and mix it, mute it and so on. Balance, gain, EQ, etc. can be adjusted for each track. Then unmute all and make final mix adjustments.
Save the sequence audio as a new file, as instructed above, for a back up. After saving the project with a unique # or name, the audio tracks optionally can be replaced with the mix.
Then use SPLIT SCREEN to assign video tracks to quadrants A, B, C and D.
If resolutions are high, with big file sizes, check out Proxy Editing. This can be worth the conversion time, especially on low-powered systems. Essentially that would be pre-sizing, per your thought, but without the need for substitution of the originals at export.
With the method described, just now created a twelve-second 2 x 2 grid quad-split on a low-end I3 PC with a conventional hard drive.
Source resolutions varied: 1024 x 576, 720 x 480, 634 x 352 and 1280 x 720; formats were AVI and WMV.
Export at 1280 x 720 took thirty-seconds and looked great.