Hello and welcome to just answer. Let me see if I can help you with this. You cannot simply disable the factory alarm on this vehicle unless you replace the PCM (engine computer) and BCM (Body Control Module). And if you do, you will need to install a BCM that does not have the security option. The 2 work together and make up the vehicle's alarm system. Now before you spend the money doing this, have the shop that installed the remote start system recheck their work. They may not have programmed the remote start system properly, or did not connect it properly.
I hope this information helps, and let me know if you need further assistance with this.
The bulldog system should have an interface to work with your factory alarm, and needs to be installed and programmed.