Once you accept a quick job, the truck and trailer (all job data) get saved to your save file, which in turns loads in MP as it is reads as default data of the game.
There is no way around it, you will need to accept a job with the mod in SP and save, and then join MP. Repeat every single time.
Same as any trailer mod.
Would be best just to take a loan for a truck in your case, but career in MP is not really possible due to damages of other players/trolls.
Good luck
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Hook'IT's post in Help with modding on TruckersMP was marked as the answer