not all immigration offices will let you get an extension shortly after you arrive. Some will tell you to come back on the last week. Check with Immigration first.
You shouldn’t have a problem with Immigration. But, as you said, the airlines are supposed to deny boarding if you don’t have an onward ticket. Some airline clerks ask, some don’t. If they do ask, you can “rent” an onward ticket from a website or you can buy a refundable ticket, then cancel it after your flight.