Constellation rethinking...

Congratulations! 👍

As you allude, there are supposed to be chronometer 562s in Constellations but NO-ONE here has ever put a purported authentic one up for scrutiny by the members.

As a deciding factor, the movt serial no. above dates it to around ‘65/66 and the case and dial are from an early 60s reference

English Connies often seem to have different crowns to Swiss versions.
Whether this is because they have been changed at service or Dennison/Shackmann took the opportunity to provide crowns that are easier to use at point of sale would be difficult to prove/disprove.

Note - this 168.5004 was produced around 10 years after the 168.004 ( style) was released and around 4 years after the last ones were produced and by this time the Swiss-produced ‘straight lug’ Connies had protruding crowns (168.016/168.018)