I find it hard to believe that Bioshock Infinite should work worse than other shooters with the Steam Controller. I played the game with mouse and keyboard when it came out and it worked perfectly fine and in my experience all FPS games that work well with mouse and keyboard (no weird mouse acceleration problems etc) translate perfectly to the Steam Controller you just have to create a good standard FPS template configuration for yourself - dont rely on community configs they arent very good most of the time and also they are different for every game so you cant build muscle memory.
My basic FPS settings are : mouse on trackpad, sensitivity 10 ticks above default, acceleration off, trackball mode off, smoothing off, edge spin radius default, edge spin speed 80 % (this helps a lot with fast reliable turning, you dont have to swipe repeatedly for big turns), sensitivity vertical scale default (I prefer that over weird ratios for this setting)
gyro mouse, always on, sensitivity 3 above default, smoothing off.
Right and Left Trigger soft pull (right mouse button/ left mouse button) with very low soft pull point so you dont screw with your gyro aiming when you pull the triggers.
Maybe try these settings, Im not saying they are the best for everyone but I do very well with them even in online FPS like BF1, Overwatch, CSGo, Rainbow Six Siege etc...
Most FPS games I tried have issues with combined gamepad+ mouse mode so in those cases you should always go for full mouse keyboard mode - never settle for mouse joystick its terrible for aiming.
For the iron sight setting in Bioshock Infinite (I think its middle mouse button if I remember correctly) you could try a few different things.
Bind it to left trigger and put Plasmids on one of the grip buttons or use left trigger for plasmids and bind iron sight to right trigger soft pull and shoot to full pull with the "hipfire aggressive" option.
Either way I think its worth it to invest some time into making an FPS config template for yourself and stick to that instead of relying on community configs.