Success in a similar endeavor with USB
At my work, we did a very similar thing.
We have a camera outputting serialized video data over a differential pair. Some of our cameras use custom flex PCB or flat flex cables for the connection. Flat flex cables can be shielded, and flex PCBs can be designed so that the traces have whatever impedance you need, and this worked well for us. We also had 3.3V and 1.8V power nets and an I2C bus running over the connection, so a custom cable was practically mandatory. Note that 3.3V power and raw serialized video data are not a USB bus.
However, in a newer revision, we had the challenge of routing the cable partially out of the enclosure (where FFC and flex PCBs aren't possible for durability reasons) and had an opportunity to use a new SERDES chip that embedded the I2C with the video data. We also added a SOT-6 1.8V LDO to the camera module, so we only needed to send 3.3V across the cable. Suddenly, (OK, so this was really planned) USB cables and connectors were ideal.
USB cables and connectors (like HDMI cables and connectors) are manufactured in enormous quantities. Economies of scale make them far cheaper than custom cabling and high-speed connectors. Our product passed FCC review, was approved by our legal department, met automotive qualifications, and has been deployed in tens of thousands of vehicles worldwide. It may be important to note that the USB cable is inaccessible to the consumer behind a screwed-down case, and is not removable is therefore invisible; there can be no argument that we're claiming compliance with the USB standard.
We're a major American company supplying all the major automotive companies; this was all above-board. There are other products (as BarsMonster mentioned) which use logo and trademark-sensitive interfaces for other purposes. You may be able to use this as evidence that it's acceptable in the absence of a lawyer's input.
Check with your legal team
Even given the existence of other products which do this, you should probably have your legal team or lawyer review the situation. The issue that will trip you up is probably (IANAL) trademark compliance. The letters "HMDI" and the logo:
(note: I believe this is fair use!) are trademarked. You can't manufacture products which use these logos and descriptions unless you're approved by the organization which holds the trademark. If your product isn't sold, or if the trademark is not visible, you're also exempt. The trademark usage guidelines are described on the HDMI website:
and you'll be using the trademark on a non-compliant product as a non-member of the organization. For our USB system, no trademark was visible. For yours, the Name and Label Requirements for HDMI® Cables say:
All cables must be labeled, both on the cable itself and on the front of the cable packaging, with the appropriate Cable Name Logo.
There is some further description of labeling on the cable itself (which would make the trademark permanently visible on your product; it wasn't on ours), labeling with a wrap-around sticker (which you probably wouldn't be allowed to remove) and labeling on the connector moldings. If you get cables with labels only on the connector molding, you may be able to hide this inside your product, but then it can't be unplugged. I'm not sure if that will work for you.
The final answer in all questions like this is always "Ask a lawyer".