A220-300 Captain reported rejecting takeoff due to an engine fault and while troubleshooting the engine fault; the engines runs exceeded the recommended limitations with customers onboard.

Narrative

Aircraft X; A220; ZZZ to ZZZ1; taxiing to XXR for takeoff. Started 2nd engine during taxi. Shortly after start; got: LH ENG FAULT" - cyan EICAS. "30 - LH ENG COWL AI REDUND LOSS" - info synoptic. Coordinated with ATC to pull over and assess our options. Quick check of MEL showed no available MEL to apply to this message. Called Dispatch/Maintenance Control to discuss. Maintenance Control confirmed no applicable MEL. Decided to conference call Airbus Tech Representative; Name; for any additional insight. Name had information about this condition and suggested: shut down LH engine; restart LH engine - swaps channels; shut down LH engine 2nd time; [and]; restart LH engine - swaps back to original channel that triggered our LH ENG COWL AI message. After accomplishing these restarts our COWL AI message had cleared.We were then instructed to run up the engine to approximately 80% N2; and then manually turn the LH Wing AI ON. We did this with Maintenance Control and the Airbus Representative on our conference call. The engine was operated at approximately 82% N2 for roughly 5 seconds while the Cowl AI was operated. Everything now worked normally; and the system had been reset. Maintenance Control advised that a logbook entry would not be required since we were able to reset the EICAS message... the problems we had encountered could be considered "transient" and we were authorized to continue our flight without logbook entries to address. The entire conference call and reset took approximately 20 minutes; and we were able to initiate our takeoff roll with minimum takeoff fuel +150 lb.After we had climbed above 10K; I researched the FOM for engine runs with customers onboard and discovered that we should have limited thrust to idle power for that situation. This was my fault as I did not realize the specific details about our engine run policy with customers onboard."