The answer is: On the 424mkIII,...
Using DIRECT recording mode pretty much precludes also using BUSS mode. You'll just have to come to terms with that, I'm afraid.
However, the fix to that, would be to get an outboard 4-BUSS mixer, such as
the Tascam M-30, M-308, or others, which will enable you to use DIRECT-TO-4-TRACK-LIVE recording, while mixing your [up to 8] inputs to a 4-buss scheme, on the M-30.
An M312 and M320 would enable you to mix 12 and 20 live inputs, respectively, to a 4-buss scheme, while recording 4-DIRECT mode on the 424mkIII. Same concept as above, with the M-30, 8x4x2x2 mixer.
The reason I mention the M-30 so often, is because that's what I use. The M-30 can be had off Ebay, very cheaply, and it provides a significant boost in production power to the 424mkIII, which obviously has a few little shortcomings, despite being a very nice machine.
Remember, on the 424mkIII, using the DIRECT mode recording pretty much precludes using the BUSS mode, simultaneously, one reason being, that your DIRECT inputs are not separated from the BUSS effectively, AND, your buss-dedicated inputs are NOT picked up at all, as you have found out already.
I don't want to repeat myself more, & wear out my welcome.
