Need to cut copper pipe after removing push pull valve and replacing with sharkbite?
It sounds like do to the nature of your faucet that when the water shuts off it shuts off so abruptly that it creates a serious amount of pressure against your valve. While in theory your sharkbite should hold up to that pressure  i would be worried that over time it may get weaker. 

For piece of mind i would install quarter turn compression valves, they are not hard to install and do not require soldering. Either way you should make sure that your copper is in good condition, no burrs or scars. Use some emery cloth to burnish/clean it up to see if it in good condition and not deformed. 

You do not want anything that can damage the rubber o-ring in the sharkbite when you slide it onto the copper. 

If you decide to go with the sharkbite you may want to install water hammer arrestor's to absorb the shock when the water shuts off.