use MOSFET as a switch in saturation region
but how about this? won't this switch work?

It will work (for a short period of time) but, the MOSFET will be dissipating about 3.2 volts x 1.25 amps = 4 watts. You'll need a heatsink and this is quite wasteful of power. However, the MOSFET you linked is a 2N7000 and, a heatsink is out of the question hence it will burn in a few seconds: -

What you should be doing is choosing a MOSFET with much much lower on resistance and providing a supply of 2 volts.