User blog comment:Edwin Shade/Turing Machines and a Problem by Vel! Posed Half a Decade Ago/@comment-1605058-20171010173654/@comment-1605058-20171024081944

Given this description, I believe this function won't be too much stronger than BB(n), namely because it can be simulated by a normal TM. This is not so obvious, because one would expect that before a single step of big TM we would have to simulate a step of infinitely many small TMs. However, at all times, we only need to simulate the small TMs corresponding to finitely many cells - the ones which have been reached so far by the big TM - and when we get to reach a new one, we can just take the time to simulate it to the point where it should be at the time (so kind of a ).

Given the above, there is going to be a computable function f(n) such that the analogue for busy beaver for your machines is bounded by BB(f(n)), so not even close to BB(BB(n)). All the same applies to the iterated construction, and probably even the diagonalized ones.