There's something up with your terminology. There isn't even such a thing as integrated counter depth, though I take your meaning, so to make sure we're talking about the same things:

• Full depth fridges are free standing. The depth depends on things like how much air space you need behind and if you're counting the handle, but they're in the neighborhood of 33" deep.
• Counter depth fridges are free standing. The same caveats apply about the depth. Most are around 26-27" deep.
• Built-in are, um, built in. Attached to the walls and cabinets. Without handles they're usually 24-25" deep.
• Integrated are a special subset of built-in. They are attached, and their panels meld right into the cabinetry so that the shape, or styling differences (e.g., different color door or larger handle) is the only giveaway that it's a fridge. You can put divided cabinet panels on them to make them disappear entirely.
• Panel ready means accepts custom door panels and hardware. It is available in some models of all kinds of fridges, though most models of built-ins have a panel ready option.

For the antipasto tray, many people keep the old fridge in the garage.

You can make a full depth fridge look built in, or at least counter depth by recessing it into the wall. There may be a few that can even been made to look integrated, but there might be compromises to the door if you do. I'm not sure if you meant integrated, however, since counter depth aren't... Maybe you meant paneled?

They are integrated to fit into standard cabinetry, so they're all gonna be less than counter depth...

I have Miele integrated columns. One of the things I liked best (and still do) is that the door bins will hold a half gallon carton but nothing wider. That means the shelves hold enough that you can actually lose stuff in the back. Thermador has the same box, but I don't remember if their shelving is different or not. The funny thing is, while some makers accommodate demand by putting in gallon jug holders, the door is the worst part of the fridge for storing the milk! It gets too warm. I have water and condiments on the door. They don't mind. :)

The main thing to keep in mind is that integrated is usually based on a 25" depth with doors. Some cabinet companies cut costs by making their boxes 24" with the doors on. Inset cabinets would be that way too. You can shim your boxes forward if you need to compensate. it's just something to be well aware of before you go for installation.

Do take a tape measure when you shop. You might find that the fridge you like will actually accommodate your antipasta tray. My only issue with getting trays in mine is where to put the the displaced items to clear the shelf. :)

I did once have a slightly too deep tray, but I made sure the shelf lined up with open space on the door above the jars (I think I adjusted something up or down -- the Miele has dead easy adjustments of under an inch each all the way on both the shelves and door bins). That gave me an extra few inches.

Integrated means it fits into the cabinets like another cabinet would and is invisible. If you want giant capacity, then you are going to have to do 2 36” modules. And maybe even add a 24” module. 25K worth of cold storage.

Just remember when ordering/preparing trays to get items that won't slide if you have to put the tray in askew instead of flat. :)

Assuming the tray fits, altogether, if the food is flat, like a deli tray, you can also put clips on a shelf with room under it, and string an X of bungees across to hold your tray mid-air. :)