Honestly, I dont think you'd need to get out of the country.
If you apply through Paper Application, it takes 80 days. You will definitely get an AOR before the officer even reviews your application. To make this work 100%, send in the paper application just 14 days before your WP expires. But send it through Xpress post or Next Day Delivery. So incase, some screw up happens in Postal, you still have time to send another application before your WP expires
Your 80 day implied status will start from the day CIC receives your application. From current trends, your AOR will be in your hand (and in GCMS) by 1st June - well before the officer opens your file for BOWP.
Regarding proof. I applied online. For me it was the online confirmation. In your case, I believe the courier receipt and your fee payment receipt will be the only proof.