Thurrock were punished for fielding ineligible player Joel Barnett who it was later discovered was banned for not paying a £16 fine two years ago.
The money was owed to a local Yorkshire FA after Wakefield Sunday side Wilton folded and their outstanding fines were shared among the staff.
Thurrock managed to secure safety in the Isthmian Premier Division by a single point last season, only to be told they would be docked three, almost a month after the season had ended.
The Fleet failed in appeals against the decision - with the Isthmian League and the FA - and suggested last week they would take their case to CAS.
But, with that process likely to take up to 109 days and to jeopardise the start of the league's season, Thurrock have opted to issue a notice of arbitration, which will see an independent panel assess the club's case.
The panel's decision, which is expected within three to four weeks, is binding and Thurrock would not be able to launch a subsequent challenge with CAS.
Key dates in the Thurrock saga
•20 April - Ryman Premier Division season ends with Thurrock 19th on 41 points
•7 May - Thurrock are deducted three points for fielding Joel Barnett, a player ineligible because he owed the West Riding FA £16, and relegated as a consequence
•24 May - Thurrock lodge an appeal with both the FA and Ryman League
•10 July - The appeal is rejected