Two wall switches and remote ceiling fan/light questions
You'll want to switch the fan switch out for a wallbox-type remote setup
The thing with the remotes that come with fans is they don't play nice with anything else that wants to control the fan; a few manufacturers provide battery-powered wall-mount remotes, but anything beyond that is out of the question.  Since you have separate wall switches with separate gangs for the fan and light, though, you have some flexibility in how you do this, and my recommendation would be a Lutron Caseta fan controller.  This mounts in the wallbox instead of the current switch for the fan, and provides both wall-switch and remote control of the fan, including speed switching, without the need for any further parts.  You'll have to take the old Harbor Breeze remote receiver out of the picture if you go this path though.