You shouldn't do it, it isn't that it is illegal, the issue is you are claiming (tacit as it may be) that you are incorporated when you aren't. This opens up a slew of legalities when people doing business with you assume you are when you aren't. If things go sideways they could claim you weren't acting honestly which can get them out of paying you, get out of contracts with you... all sorts of things.
I've no firsthand knowledge, but I'd be surprised if there wasn't laws surrounding holding yourself out as incorporated when you're not. Probably for exactly the reason you're doing - trying to make it look like you're incorporated.
If you're not incorporated, just go do it. It's substantially less expensive and work to become incorporated than many would think. Years ago I set up a US incorporation online, it was easier than buying junk on ebay.
In my line of work most people aren't incorporated either - but I am. It's really not that hard to do.
You should definitely consult with a lawyer and or tax preparer before rushing into incorporating- there can be a lot of tax issues. For example, many states require you to file a tax return until you formally dissolve the company (as does the Federal government). And California hits you with a minimum of $800/year in taxes, whether or not your company makes a profit.
Well, I don't know if it's quite that fancy, but a call to your accountant wouldn't hurt. I don't think an independent needs to spend a whack of consultancy money figuring all this stuff out.
I incorporated in the US from outside the US, did it all online. I had to pay the fee to do that, plus I think I had to pay a lawyer a fee to act as a local physical representative or something. And yes, if I recall, I filed an income tax return in the US every year as well.
But it wasn't onerous. I'd say I maybe spent $200-$1000 a year on it (and probably closer to the $200 mark). And I only did it because I had a lot of US clients at the time so I figured it was appropriate and seemly to have a US inc. and address. In short, I didn't have any problem setting up a US inc. for little more than vanity purposes.
You can put MyBusinessName.com™ without owning the ™ ..unless someone else does ..but you won't "own" it unless you register it as a trademark ..but if you use it ™ for long enough ..you can "sort of claim it" as long as no one else registers it ..
Even simpler would be to register it as a ™
All of this depends on your jurisdiction..and I'm not a lawyer etc YMMV..