How do I increase photo resolution for printing?

You cannot change the resolution of an image. The resolution of an image is entirely dependent on the number of pixels it has. If you want to provide an image that will print, for instance, an 8 x 10" image at 300 pixels per inch then that image needs to have 2400 x 3000 pixels (8 X 300 = 2400 10 X 300 = 3000). Changing that setting in a file isn't going to do anything to increase or decrease the quality of the print. If you have the appropriate number of pixels then you will be able to obtain a print at the desired resolution regardless of what the setting is in the file.

Work in total pixels, the resoution tag (300PPI not DPI) is really meaninless. 

I wrote my first article for a magazine on Resolution way back in 1998 and I see it's still a topic of confusion today. FWIW, the article is archived here:
http://digitaldog.net/files/Resolution.pdf
Let's start here with some caveats and why I suggest always working in pixels. 
1. Digital images don't have any size other than the space they take up on some storage media. This size varies by many attributes even if the document has the same number of pixels: bit depth, layers, file type and possible compression, color space. It's not worth even considering this size due to so many differences. Digital images therefore should be considered in pixel density. And for this discussion I'm going to limit this to one axis (let's say the long axix) and the image is 1000 pixels. 
2. An analogy is necessary to discuss the resolution tag in digital images. If I'm 6 feet tall and every stride I take is 3 feet, and my friend is 5 feet tall and every stride he takes is 2 feet, when we both walk exactly 1 mile, we walked exactly a mile. That I walked with less strides (resolution) doesn't change that I walked exactly 1 mile (pixels). 
3. The resolution tag places no role in the 1000 pixel document in this respect: 1000 pixels at 100PPI and 1000 pixels at 100PPI are the same: 1000 pixels. In fact you can take a document that has 1000 pixles with a resolution tag of 100PPI, duplicate it and change the resolution to 1000PPI and the two are identical other than for metadata such as this resolution tag. And of course metadata like date/time the document was created and so forth. The two documents are 1000 pixels and the tag has no role and does nothing at this time. Set it for anything you want, as often as you want, it's the same digital image at this point. 
For all intent and purposes, the resolution tag plays no role. The number of pixels does. But wait you say, "I want to output the 1000 pixel image". To a print or on screen. OK, now we have a new size to consider! Let's work with a print. Computers are not too smart, they have no idea what you wish for a print size until you tell it. They do know you have 1000 pixels to use to make the print. What size print do you want? The answer comes about when you divide up the pixels you currently have (more about what you might have later) for this print. Now size can be inches, feet, meters, miles, CM, MM you get the point. Let's stick with inches for this story. You have 1000 pixels and the resolution tag is set to 100PPI. You simply need to understand simple math (division) or have a calculator once you accept you have 1000 pixels. At 100PPI (the tag), a print could (repeat could be), 10 inches. If the tag is 1000PPI, you're going to end up with 1 inch if you allow the computer to provide that division of your pixels. If the resolution tag is 23PPI, the size would be 43.4783 inches (here's where a calculator is useful). It's not if the tag is in MM or CM, or you alter the tag value. But in every case, the data is 1000 pixels. That is the critical number to know about first. The other number can always be changed so software can at this point understand a potential size for output. 
OK, so now Lightroom (or Photoshop or anything else) comes into play. And you ask for that 1000 pixel document to be output to 10 inches at 200PPI. What's a computer and software to do? You don't have 2000 pixels. So the software will interpolate and add more pixels out of thin air so to speak. Or you could reduce the number of pixels with interpolation. This is where Robert got a big flummoxed. He said "I didn't interpolate, LR did". Of course it did! If you ask for output that requires 2000 pixels and you only have 1000 pixels, AND you give the software permission to make more pixels, it will. It will interpolate. It interpolates because you told it to interpolate and make more pixels due to the size relationship with the current tag. 
Now to the deal with sharpening. Until you print the image, it's still 1000 pixels. The PPI tag is moot. But you asked in the print module for a print at a specific size. And guess what, LR sharpens based on what it knows about the number of pixels (current or what you might, repeat might interpolate) and the size you asked for. Output sharpening is output resolution and size specific. So that tag is NOW used. But you still have and have always had a 1000 pixel document san's permission to interpolate adding or removing pixels. 
Work in pixels. Have a calculator nearby if necessary. Pretty much ignore the resolution tag until, if, you need to output that data and you require a specific output size or sharpening in the case of LR. Understand you can allow software to interpolate BASED on the resolution tag. If the tag is 100PPI and you tell LR you want 10 inches, the results are quite different than if the tag is 1000PPI and you tell LR you want 10 inches. You are in control. The software only looks at the tag if and when you tell it to look and use that tag to produce some size with the pixels you have.  

It sounds like you have a very weak understanding of your client's needs, and the first thing you should do is come to better understanding of what (s)he is asking for.

Lightroom Classic does not control DPI (dots per inch). DPI is not a characteristic of a digital image.

Lightroom Classic can control PPI (pixels per inch) which is different, and is sometimes called resolution. Is this what you are referring to? If so, then when you export a photo, you click on the "Image Sizing" box, you specify a resolution of 300 pixels per inch, then you specify the width and height in inches or centimeters, and then you export.