How can I replace a riveted doorknob faceplate?
Go with plan B. The larger strike looks like either of two things, a reinforcement for a damaged mount for the strike bolt. The door slab looks ever so slightly thicker at the strike location, like the screws somehow split the door at the strike. Or the door originally had a mortise lock set and this part serves as an adapter so to get a standard knob to work in the large cut out for a mortise lock. 

I really think it is the former, not the latter. Either way you need to at least reuse it, or replace it if you can find one at the big box stores. I have seen them before. It should be where all the replacement parts for old style doors are located, I do not know what it is called.  You can remove it temporarily if it helps you drill out the pop rivets. Also, it may help the store guy see what you need if you want a new one anyway just to freshen up the look to go with the new knob.
 You can drill out the pop rivets which MAY be 3/16" but drill first with a 1/8" bit, and if it doesn't fall apart, then use the 3/16". After that you can purchase a standard bore latch set and new pop rivets and get it all back together again.

For what its worth, you could do away with reassembling it with pop rivets and go with 8 gauge flathead machine screws in a bright brass finish to match everything, and use "stop nuts" to hold it together. Get a couple of screws to replace the drywall screws to balance it all out.

If you feel adventurous and you have worked with chisels before and you are adept at it, you could add a "Dutchman" of substantial depth to do away with the secondary metal plate, and get it closer to the way it is meant to be