Hello bennage,
Strange that auth works for GET’ing and not POST’ing.
I think I would start by looking at the OAuth redirect URI in zapier dashboard for your zapier app, and then check that is added to the OAuth redirect URIs in your Freeagent app in the developers dashboard.
Next you may need to look at the query string on the Redirect URI when attempting to post something - it may be related to this post -
I do a similar thing with bills and zapier so it should work. Let us know how you get on you may need to post the string that is failing authentication with the private bits censored.