Thank you guys for the overwhelming flood of comments!
Some were really helpful, but I think, my point was not understood entirely.

I use the described combination of film, exposure and developing for a while and am happy with the results.
I do not seek to deviate from:
film (I like the look and multi purpose qualities of TX),
exposure (I cannot raise exposure unfortunately but would actually have less exposure ideally as I am at the bottom of shutter speeds already),
developer (I like to keep things simple and D-76 simply works perfectly well - I can easily source it, store it and mix it very economically)

My intent was to gain more knowledge of which slight changes to my process might result in improvements, small or not I am happy to learn and tweak the process.
I cannot believe that changes to my agitation will have any visible influence.

Unfortunately I cannot switch to 3200 ISO films - the costs are absolutely prohibitive (+ sourcing them locally in 120 is impossible).
I do not have more exposure (and frankly do not care about better tones at ISO 200, as it is not available to me for when I need ISO 3200).
When I want better tones and have more exposure, I simply use Tri-X at 320 or just shoot digital.

I read a side note about agitation and dilution with D-76, even fixing time having potential influences on detail and grain here:

…and simply wanted to hear about these minuscule effects from the experts, so I can improve upon the developing I do now.