Can you try to have a civil discussion were on paper you both write out what mentally stable means. Go into as many specifics as possible. Would silent treatment count, not going out and doing things with people, yelling, breaking things. Then post it somewhere in sight like on the fridge. Perhaps allowing for a warning every so often. Also try to determine the type of pet that is allowable, perhaps she would feel more comfortable with a caged pet like a hamster.

If this isn't possible, give it your best shot. Assuming things will end badly is something I do all the time and I'm more often incorrect then not.

If at the end of the month all of that fails then perhaps try volunteering at a pet shelter. If that's not possible make friends with a neighbor so you can co-own a pet. Growing up I walked the neighbors dogs for free and spent a great deal of time playing with them.

Best of luck.