Working with inductive loads
A detailed answer to the SSR question follows a more complete look at what causes contractors to fail, but the short answer is that the required SSR will be rated 300A-400A and 370V. The SSR plus heat-sinking and MOV snubbers will be around 2x the cost of a reasonably good 40A contactor and the wiring and mechanical layout will required some skill to install.
An SSR is typically contemplated when a relay or contactor has failed, and it’s not always the best approach to improving the reliability of an existing machine. Reasonably good fully derated contactor (e.g. 40A contactor for a 12A FLA compressor) should last &gt;100,000 cycles +/- 50%.
If a system has a (truly excessive) 10 cycles an hour (6 minutes between starts) that’s around 80,000 cycles a year. At that point an SSD makes sense, but to realize increased system reliability requires upgrading other components for motor starting and to run the condenser fan after compressor shut off to prevent starting into head pressure. If the 2 ton machine has run cycles that are always more than 10 minutes and usually longer, and has a 20A contactor or just an “ice cube” relay, then an upgraded electromechanical switch is probably a better next-step.
Verifying or upgrading the motor’s starting system is also appropriate.
A failing start or run capacitor will cause excessive starting current that over-stresses the power contacts.  Other causes include wiring or shutoff switch with a large voltage droop, or a weak 24v control transformer not closing the contacts rapidly during the turn on transient causing them to erode rapidly.
Good industrial control transformers are larger and cost a little more than low cost 24v 40VA control transformers, but relays snap closed quickly. If an intermediate relay bounces or any other device causes the power-on signal to flutter then the power contacts chatter closed instead of snapping closed and they erode rapidly.
Repairing a failing upstream control is preferable to immediately installing an on-delay timer set to 1 sec to de-bounce the relay drive, but diagnosing the problem requires an oscilloscope and a particular model of relay may simply bounce, so it’s more cost effective to clean up the drive signal with an on-delay until it actually fails.
The compressor motor will start most efficiently if it’s started by a starting relay that senses the voltage (or current) in the start winding along with the correct size start capacitor. Researching the correct pick-up and drop-out voltage (or current) and starting capacitor for the compressor is required.
There are a limited number of “good” choices for designing a given size motor, so a similar model from the same manufacturer is likely to use the same starting components. If service information gives the same DC resistance for the windings and the run capacitor is the same it’s very likely to use the same starting components. Some/many 3-wire electronic aftermarket start relays with adjustable pickup/dropout voltage don’t perform as expected and function as simple time-out start relays, as will most start relays that say “universal”, caveat emptor.
A simple time based start device like a PTC or a 2 wire hard start is on for the longest possible start so it’s almost always on longer than it should be which wastes energy and is hard on both the motor and the contactor.
An oversized 2-wire hard-start device is particularly punishing. So - make sure what’s needed isn’t a motor capacitor(s), a better control voltage transformer, an on-delay to clean up the control drive, a new power shut-off or power feed “whip”, a “real” motor start relay, or tightening some wiring connection (look for the one that looks like it was cooked ‘well-done’). If there isn’t a simple cause for the relay component failure, it’s simpler and cheaper to upgrade to a 40A contactor than an SSD.
An SSR for a conventional 2 ton 12A (full load) air conditioner will be turning on to spin up the compressor’s internal split phase motor and the condenser fan’s split phase motor. The SSR needs to survive the motor turn on current transient, typically between 8 and 12 times the full load current. This transient persists for only a fraction of a second, but will damage an under-sized SSR over time.
Of greater concern is the stalled motor transient that is the same stalled rotor  current for 3-5 seconds until the motor protection device snaps open. These stalled compressor events do occur for a variety of reasons, so instead of using 50% of the 144A peak current (72A) for the SSR current rating, the entire 144A rating is used. These events are more likely to occur at either low outdoor temperature due to liquid migration, or very high outdoor ambient temperature when a simple restart time control may not be long enough to get the condenser pressure down far enough to spin up an aging machine.  The SSR must also be derated for high ambient temperature, 50% for most locations, 70% in the warmest areas, so now the target device is either 288A (300A) or a 400A device.
The price of this SSR component should be very close to price of an electromagnetic contactor, but heatsinking and inductive spike clamping are needed. The heat sinking should conform to the SSR data sheet. The MOV should be chosen with guaranteed calming voltage at 30% over the peak line voltage, 1.414 x 1.3 x (line voltage).
If the air conditioner is using a variable speed compressor motor with an inverter drive, a different set of design considerations apply, and adding some inductance to limit the current inrush is required, but this is an answer, not a textbook, so only a retrofit for a conventional machine is considered. Likewise the compressor transients are assumed to be much larger than the condenser fan and any condenser fan speed controls are assumed to be benign speed controls, if a bang-bang control is used e.g. a pressure switch, then additional MOV spike control at the fan may be needed.