As mentioned, it will work with the rear derailleur, and a road FD should be pretty affordable. But, if your shifters and brake levers are integrated, they are probably set up for linear pull brakes.

For some brake levers that are adaptable for both, the difference is the point where the cable connects to the lever.

I have seen brakes that work on both, that basically have a method that allows for using either attachment point.

I can't find a diagram online, but depending on the lever, there might be a way to get your brake levers to work with caliper brakes... not an easy way, but maybe a way.

I will keep looking, but if you can take a picture of your brake lever with the lever actuated so we can see the attachment point, I may have a suggestion.