Undecidability of whether a given TM halts or only has loops which can be detected by some TM
(too long for a comment)

Here's my attempt at an unambiguous version of what you are trying to get across.


  Is it the case that for every total Turing machine M,
  
  if[
  for all $\langle d,n\rangle$ in $\{0,1,2,3,...\}^2$, $\:$ if M accepts $\langle d,n\rangle$ then
  
  $d$ describes a Turing machine that enters a loop on input $n$,
  ]then[
  there is a many-one reduction from the Halting Problem
  
  to the set of $d$ in $\{0,1,2,3,...\}$ such that
  [
  $\hspace{.01 in}d$ describes a total Turing machine or $\big[\big[$for all $n$ in $\{0,1,2,3,...\}$,
  
  if $d$ enters a loop on input $n$ then M accepts $\langle d,n\rangle\big]$ and there exists such an $n\big]$
  ]?