How to determine correct mosfet for application?
If you want to control the FET with the output of an ATTiny, you want a logic-level MOSFET, which are designed to have their gates driven at lower voltages that microcontrollers would operate at (e.g. 5V).
The logic-level equivalent to your IRF540 is the IRL540. The IRL540 is well within your parameters for current and voltage capabilities, and it has a fairly low Rds(on) of 0.07ohms at 5V Vgs and 0.11ohms 4V Vgs. This means you should have no trouble running it without a heatsink even if you only drive it at 4 volts—\$P = I^2*R\$ is less than half a watt of heat dissipation, which shouldn't be too much of a problem for the FET, but it might get a bit toasty...a small heatsink certainly couldn't hurt and is very cheap.
You could look around at other logic level FETs to see if you can find one with a lower Rds(on) if you determine that's too hot to run without a heatsink.
Edit: Even better would be the IRLZ44, with an even lower Rds(on). Driving it at 5V, power dissipated is \$2^2*0.028 = .112W\$. RthJA is 62 C/W, so \$0.112*62 \approx 7℃\$ increase above ambient. If your room temperature is a toasty 25C, your FET will heat up to 32C. Quite hot, but well within the operating parameters, and certainly better than the 50C+ that the IRL540 would reach.