Do you mean to use the define tool rather then project? I just took this literally and did both, just to make sure.
If I define WGS84 from TM65 then it is just off the west coast of Africa. I can't seem to preform the project without selecting one of the transformation options? I believe I did manage to do it when I was trouble shooting this a day or so ago and it was out as well.
But could that be a clue to the problem?
I carried out the TM65 to TM75 then TM75 to WGS 84 using the transformation you suggested and it is still out by about 9 meters. The grid files I have in my system are tm75_etrs89 (Ireland folder) and OSTN02_NTv2 (UK) folder.
The area I'm looking at is in the North of Ireland. So it is on the same land mass and grid as the Irish Republic.
I'm using the ESRI aerial imagery basemaps as my target comparison.
S.