Designing Switching Circuit for Raspberry Pi 3 Project
Few points.


A switch is a short between two points. To measure current through a switch, just place the multimeter in parallel. The multimeter will act like the switch making it active. In most cases. No need to desolder the switch.
Plenty of 3.3V relays out there. But you never want to power a relay directly from a GPIO anyway.
You know the ICE that you need (35mA), and the max current of the RPI (16mA). So any transistor with a IBE * hFE that is equal or higher to your ICE will work. In this case, any common "jellybean" transistor like the 2n3904 or 2n2222 will work. Don't forget the base resistor.
But it's pointless. You can just as easily use the transistor to directly control the switch. Place the C-E across the switch and your done. No relay needed. If the button is active high, a PNP would work instead, with reversed logic (gpio low enables the transistor).