Seriously, the only ugly injury I had to wear to work was a nasty busted knuckle I got when I was shadowboxing at home and hit a wall (yes, alcohol was probably involved...)
I had to explain to like a dozen people in the office why I had this big bandage on my knuckle. As embarrassing as the truth was I told it anyways and no one said anything negative about it, at least not to my face.
I've worked in HR and if you're concerned about this affecting your work I'd do the following: (Note: Outside the US these rules might not apply)
1) Don't let it affect your work. As long as you don't deal with customers/clients/business partners face-to-face you should be okay.
2) Don't make a big deal out of it. If there's a problem make sure it's someone else's problem and not yours. You're just a guy who trains in Muay Thai and this is just a part of training, like runners getting shin splints. Not a big deal.
3) If the boss has a problem that the two of you can't work out talk to someone in HR (if you have an HR department). As long as your injuries aren't affecting your work this should get swept under the rug.
The only issues I could see besides this is if you are consistently coming in with a face that looks like raw hamburger and it's bothering other people. That might lead to you getting terminated, just to keep the peace.
Or if you start having excessive health-insurance claims that might become an issue to.
In any case if you end up getting terminated for sports-related injuries you will be able to get unemployment compensation, and you might still get a good reference to.