how to correctly measure print size & distance from web?

You don't need dpi of anything of this sort.

Because you have a box (t-shirt in this case) of fixed and known size and position.

When a user moves an element inside the box you can calculate the element's position and size relative to the box.

And all you need to do is something like:

scale = box.width_in_inches / box.width_in_pixels
el.x_in_inches      = (el.x_in_pixels - box.x_in_pixels)*scale
el.y_in_inches      = (el.y_in_pixels - box.y_in_pixels)*scale
el.width_in_inches  = (el.width_in_pixels              )*scale
el.height_in_inches = (el.height_in_pixels             )*scale

box.width_in_inches can change when user selects t-shirt of a different size.
box.width_in_pixels will change when user zooms the page.
But all *_in_pixels values will change respectively.

Update: Answering to your comments:

1. Maybe your're getting wrong measurements in your code because they are wrong from the beginning: the ratio of t-shirt's width and height

   1. in inches: 22 / 30.5 = 0.72...
   2. in pixels of the picture: 452 / 548 = 0,8248... (or on the page in my browser: 1428,750 / 1732,190 = 0,8248...)

   3. in pixels with bounds you've set up: (1428,750−160) / (1732,190−100) = 0,777...

      So I had to fix it somehow. I choose to remove the bounds, so that the logo can move and stretch inside the t-shirt picture freely. And decided that t-shirt width in inches should be 30.5*(452/548) = 25,157..., not 22.

2. Then just as a sound check I set logo width to maximum and move it to the top. Now it's time for experiments. The expected results should be reasonably close to
   { x:0, y:0, w: 25,157, h: 9.07 }

3. With this function:

   // const shirt_w_in_inches = 30.5*(452/548)
   const shirt_h_in_inches = 30.5
   
   function get_measurements() {
     const pixels_shirt = shirt.getBoundingClientRect()
     const pixels_logo  = logo .getBoundingClientRect()
     const scale = shirt_h_in_inches / pixels_shirt.height
     const inches = {
       x: (pixels_logo.x - pixels_shirt.x)*scale,
       y: (pixels_logo.y - pixels_shirt.y)*scale,
       w: (pixels_logo.width             )*scale,
       h: (pixels_logo.height            )*scale,
     }
   
     console.log("pixels_shirt", pixels_shirt)
     console.log("pixels_logo" , pixels_logo)
     console.log("inches"      , inches)
   
     return inches
   }
   

   I find them quite reasonable.

   The worst case I was able to find with an maximum error of around 1% of t-shirt width is when I set a 500% zoom, and refresh the page, and then if I resize my browser to 25% of initial window size (and I couldn't set it less). In this case I get these measurements:
   {x: 0.153..., y: 0.172..., w: 24.920..., h: 8.948...}

   But if I add inches.width and inches.x I will get 25.073 - which gives 0.3% - an average error I've observed.

4. So you can ask: "What about x and y? They are never 0!".
   I'm sorry to break it to you but it looks like an issue with Moveable you're using. If I use this function:

   function set_full_size() {
     const pixels_shirt = shirt.getBoundingClientRect()
     const pixels_logo  = shirt.getBoundingClientRect()
     logo.style.left    = shirt.clientLeft  + 'px'
     logo.style.top     = shirt.clientTop   + 'px'
     logo.style.width   = shirt.clientWidth + 'px'
     logo.style.height  = shirt.clientHeight*pixels_logo.height/pixels_logo.clientWidth + 'px'
   }
   

   It not only sets zeros for x and y, but also you'll get a smaller error for width. Probably that's because of controls the Moveable puts around logo.