What are the easiest ways to deal with AI ships when playing as the Mantis Cruiser?

One option is to look for the Emergency Respirators augment. It's reasonably cheap, and halves the damage your crew take from lack of oxygen, meaning they can last a reasonable amount of time in an oxygen-free environment. You shouldn't be too worried about "playing leapfrog" with your crew, since whenever you send in boarders they're always going to take damage from something; suffocation is just a different cause. Indeed, the Fregatidae-class Federation Cruiser comes with this augment and a teleporter as standard, and I've successfully beamed Zoltan aboard oxygen-free ships and brought them back alive. That particular ship, however, is more than slightly insane, so don't be too eager to replicate its strategies.

Also, following 5pike's comment, the reconstructive teleport augment can be immensely useful here by completely healing your boarding party every time they beam somewhere, meaning that the usual problem associated with having a clonebay is largely nullified.

You could also start looking for some Lanius crewmen to add to your boarding party, who will of course be completely immune to the effects of low oxygen.

Finally, you might want to look into picking up a clonebay. Rather than bringing back suffocating boarders, just let them die and clone some more!