Just so you know pop riveting is a PITA. It is difficult to get right and even with practice you are going to screw it up sometimes. For this reason most places that put in alot of rivets use pneumatic riveters.
I am here to help, first off don’t panic. You can and will do this. Flexible materials like plastic are hard to rivet properly what you need is practice on something else that is similar in thickness. I also echo the washer thing. You may be using the wrong rivet not diameter but shaft length, this is the length of the rivet body not the nail thingy.
Go to HD or similar and buy the variety pack with washers, also look around for something cheap and about the same thickness to practice on. If you haven’t already done this.
Are you using the right size die in the riveter, that is the usually chrome part on the riveter, they usually come with a few different dies and a little wrench to exchange them. This is where the nail looking thingy goes in.
When riveting you must keep the riveter straight in relation to the rivet and material being riveted. When I rivet with washers I do it upside down to keep the washer in place and as a bonus you can see the rivet and make sure it is expanding properly.
It usually takes two or three pulls to snap the shaft. You must, after each pull, reset the die head on the face of the rivet, meaning release the handle and get a new grip on the shaft flush with the rivet.
Pop rivets tend to break close to the face instead of leaving a hole like pneumatics tend to do, this is just the nature of the beast. If you are very brave you can drill the shaft back a bit but be careful tis easy to drill too far.
Practice riveting and drilling them out til you get the hang of it, you will. Use the same diameter drill bit as the rivet when drilling out.
Need more help PM me, I am happy to help
Capt