That's what I'm getting. Sadly it's not fully integrated

Anything "counter depth" can have cabinets built around it to look "integrated" but it will never truly be a built in. It can be made to look very nice though. You'll end up spending a decent amount on the cabinetry with this setup though. Have you ever considered using fridge drawers with a built in icemaker for the bar area? It can save a lot of space and really looks sharp.

As far as I know, the Thermador (including those made by Thermador, like Miele) and a subsequent Subzero are the only completely integrated models. I'm not sure if Liebherr might have one.

Add Gaggenau to the list.

Gaggenau is the same family as Thermador -- fancier trim out, so more money instead of less. So yes, there are some others, but I don't think you will find fully integrated at a lower price point. Your best bet would be to look for a floor model being sold, if you have some flexibility on timing.

Liebherr makes a pretty much integrated fridge, except you can't cover up the toe kick. You can cheat on this with Velcro'd feet at the edges.

Bosch makes a fully integrated.

Blomberg makes one; I have no knowledge of this brand (or ever heard of it before) but in my quest for an affordable fully integrated refrigerator this is the only other brand I have come across other than those already listed. In the pictures I have seen it appears to be very similar to a liebherr.

Here is a link that might be useful: Blomberg Refrigerator