Is there a way to keep large plates together more efficiently?
This isn't my personal experience, but I have seen and heard people use rubber hammers for fixing larger plates onto baseplates, which shouldn't be all too different from fixing them onto normal plates.

Those are a classic homeworking tool, commonly used pretty much for the "real world" equivalent of what you're doing, fixing floor tiles and cobblestones into place. They should be soft enough when applying force over a flat area as long as you don't hammer away like a madman. For additional protection you could possibly also put a piece of cloth over the plates. It's probably still more effective and lesss straining than using your hands to hammer or your thumbs to press.

In general though, it's also not a bad idea to try and avoid sticking too many big plates onto each other, as even if you get them pressed together really tightly, it can lead to bending of the entire model when done over a larger area, which is especially noticeable when having a flat ground stand that then...isn't flat anymore. Some of the "thicker" Architecture sets like Trafalgar Square or Robie House exhibit this problem. Try to thin it out when you can or use baseplates for the bottom layer, which are a lot more flexible in themselves.