How would the FBO have filled out the eAPIS report and obtained landing rights for you? The FBO can certainly help in updating customs on your ETA, but I don't see how they could do anything for you beyond that.
You as the PIC must ensure that CBP has given you landing rights before departing. Period.
All the information you need is here:
U.S. Customs and Border Protection - Travel
I would suggest that until you become more experienced at international ops you retain a flight planning and handling company to assist you. $5000 may seem like a lot but it's a pretty cheap lesson in my opinion. Consider how much fun 3 days in a Mexican jail would be in comparison.