I doubt that getting to the pipes would be of any help, anyway. Could it be that the hole you see is where the set screw should be but the handle is held firm by corrosion and/or a tight fit?
Is this a single control handle unit or are there two separate valves?

It's a new single-handle Delta Monitor. I just bought the house in April. It was a flip....

It does look like there should be something there. I might try a plumbing supply store or something for a set screw, then.

Sounds like the set screw requires a hex wrench.

I have a drip in the bathtub since the house was built. My neighbor did also. She called the plumber when the house was new and it still leaked after he fixed it. They said something about a bit of something stuck inside was the problem. That's true with mine I think. I find that if I turn it on and off a couple of times it doesn't drip any more.

I removed the handle with an allen wrench but there are no proper washers, just a plastic disc and other plastic parts. Now what?

http://www.masterplumber.net/delta/monitor.htm

Remove the cartridge and check for debris in the valve body and damage on the sealing faces.

These are cartridge faucets and do not use old style washers.