When you hold shares short, your broker can "call" the loan at any time and make you purchase shares in the open market to cover the short. What would probably happen in this case if you exercise the put is that you would become short the shares and if the shares were too hard to borrow, the broker would buy the shares on your account to cover the short. That is just a guess on my part, and it's also possible that if the shares were hard to acquire, but not impossible, the broker might allow you to remain short. It would be worth a try if you did not have much premium in the option.
I have used that strategy before, not because the shares were hard to borrow, but because the broker policy prohibited shorting stocks with a share price under a dollar. For one stock that was under a dollar, I purchased near-term, in-the-money puts, then exercised them. That meant that I was now short the shares. The broker allowed me to remain short for the shares. Of course this is the type of thing that might vary from broker to broker.