I got a duel sport 1 a few months ago and have been enjoying it very much but the one problem i keep running into is the breaks which are mechanical disk breaks by default. I've found a new set of Hydraulic disk breaks that I would like to install however i don't want to have to replace the shifter. I was planning on removing the break lever and cable from the shifter housing and installing the new set of breaks apart from the shifter. I have the technical knowledge and skill to do this but want to make sure it isn't stupid or that i'm missing something obvious before I start. Would this be a viable solution to install hydraulic breaks on a duel shift/mechanical break housing? Please note that i'm trying to work with a low budget so replacing the entire shifter isn't really an option.

Shift/Break housing: Shimano ST-EF500
Hydraulic break replacement: Shimano Altus M315
Bike: Trek Duel Sport 1