What's the rest of your team? 

Yes, please show us your champ roster, full team and builds! Would love to help out with the most informed opinions. :)

Should  never  be  bulwark  from  my  use  of  the  man

I  currently  have  Rowan  as  my  stun  Target,  but  I  need  a  second  one  so  that  I  don't  have  to  keep  changing  gear  on  her  for  my  various  Budget  Unkillabke  teams.  Not  sure  when  I  will  get  a  second  Maneater...

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I have no experience with unkillable so I'll refrain from commenting on that part but - I can't see any way in which Rugnor works on a CB team. His passive is going to completely throw off your rotations.

His passive only reduces the cooldown of his A2 and has no direct effect on the turnmeter in any way. His A2 has a turnmeter boost, but you can disable this skill and use Rugnor just for the combination of leech (A1), and decrease def + weaken (A3). 

I guess you need the leech debuff, because the damage from the stun hit is eaten with only one Maneater and the hp-bar has to be filled up before the next stun. 

My  unkillable  team  is  tower  warcaster  shipi  anax  skullcrusher  As  long  as  skullcrusher  has  his  counter  attack  up  on  the  other  champs  he  will  always  be  the  stun  target  so  he  wont  trigger  counter  attacks  i  hope this  helps  if  he  is  fully  booked  he  will  only  have  the  counter  attaxk  down  for  one  turn  unless  you  use  a  reflex  set  up