If your question is "how do you know if the sand filter is capturing the bacteria bloom that you are feeding/creating by carbon dosing" then the answer is to see if your nitrates go down over the next few weeks. Carbon dosing doesn't work overnight, the smaller the tank the faster it works, but it still doesn't work overnight (I guess it could if you overdosed carbon, but you'd hurt a lot of organisms by changing the chemistry that fast, even when reducing phosphates or nitrates, too fast is bad).
Even with your sand filter and backflush method, if it is truly working as expected, you should see nitrates go down over the next few weeks. Same thing with your deep sand bed as well.
I think you are going to run into problems with the sand filter. I love the idea, it makes a lot of sense, but my guess is that you will have the classic "bioballs" problem. They work great for neutralizing ammonia into nitrite and then into nitrate, but then they become "nitrate factories". It's not that they're creating nitrate, it's that they're tons of oxygenated surface area, providing a lot of space for nitrification. Deep sand beds should work to remove this (the marinepure ceramic blocks are effectively concentrated squares of deep sand bed), but denitrification is slow. Maybe the chaeto will help and absorb the nitrates and phosphates, but I suspect your nitrites will climb as long as you are using the sand filter, unless you are washing it almost daily. I think filter socks have the same issue.
Also, you may have trouble with the deep sand bed over time too. If any detritus gets down to that sandbed, it will get trapped and eventually fill the DSB with more detritus/nutrient load than it can process, and you'll get what is classically called "old tank syndrome", where the rocks and sandbed cannot process nutrients anymore because they are nutrient-logged from years of detritus and nutrient exposure. The ceramic blocks get around this by not allowing any detritus into it (pore size too small, allegedly, nobody has been running them long enough to know, and the sample size of people with different conditions of tank running them for a long period of time is too small).
Something to consider, and I know it's a different situation, but it's somewhat similar, is that most commercial marine aquariums that have fish-only setups use reverse-flow sand filtration for their nitrification. I know you're not doing reverse-flow, but I think that you'll still have an enormous amount of aerobic surface area for nitrification, which will lead to too much nitrates.
You look handy, why not build a
DIY methanol denitrator? Basically long hose in a box that you slowly drip methanol to, which becomes anoxic, and allows bacteria to process nitrate to nitrogen gas. For reference, in the link above, the guys using it have very large heavily stocked with large fish tanks that they are running these on. It's not a tiny tank with mostly corals that are fed limited food. These guys are for real.
