Ideal component to switch 84V DC?

Use the MOSFET. Its use case is switching a DC load. If the LED strip has a lot of inductance, put a flyback diode across it.

If you drive the gate with 5 V relative to source, then the RDS(on) is specced as 0.077 ohms from the linked data sheet. The power dissipated will thus be 0.27 * 0.27 * 0.077 = 0.006 W. No need for a heatsink.

The IRL540N mosfet will most likely be the best in all areas even in terms of power dissipation. With a listed Rds(on) of 0.077 ohms (@5Vgs) it will dissipate only about 6mW at 270mA. Required gate drive from the control circuit is virtually zero. You may just want to ground the gate to ground with a high value resistor to insure it is off when required, that will also reduce potential static discharge events that could damage the mosfet.

The TIP122 and SCR will have higher on voltages due to Vce(sat) and Vt/Vf ratings, thus a higher dissipation at 270mA, perhaps as high as 0.5W, not exceedingly hot by any means but likely noticeably warm.

The opto-isolators you mentioned are also a very good way to keep the low voltage side safe.