Are there rim brakes for road bikes compatible to Shimano hydraulic brake levers / brifters?
I would pick up a pair of cyclocross-style brakes levers that are designed to mount on the flat part of the bar and run inline with the brake cables. You can also mount them to the end of the bar and just have the cable terminate at the lever. The angle might not be quite right with the bullhorns, but some levers come with an adjustment screw that lets you set the default position of the lever. I would make sure to get some like that. And then you can just use whatever bar-end shifters you want.