A ring light is most effective with the lens in the center, shot off camera it will still give ring highlights but will lose that unique ring look. Trying to handhold, fire and advance a TLR in one hand and a ring in the other is something I just do not want to think about. There are a number of reasons the TLR has sort of died out in favor of the SLR and the Rangefinder, it's advantage when it came out, are now part of it's fall from grace. If you want a waist level and love using a ring get an SLR, there are a number of them out there that have lost most of their original value and can be bought for a fraction of their original cost. A TLR was an ok walk around camera as long as you kept it simple.
For reference sake I love and have continued to shoot with a ProFoto ring light of some form for at least 25 years.
Also just for reference I have owned and/or used the following MF cameras Rollei 3.5 TLR, Mamiya C330 TLR, Bronica C SLR, Contax 645 SLR, Hasselblad 500cm and H1,2,3,4, Mamiya RZ 67, Mamiya 6, Mamiya 7, Fuji 680, Fuji GX617, Linhof Technorama 617, Holga TLR, Widelux and others. If I was to pick one up for just having fun shooting on the street, it would be a Mamiya 6,(a camera I should have never sold) or a if you could find one a Widelux (a camera I should never have returned), for most people and especially if you want a waist level viewfinder, I would suggest a Hasselblad 500cm, none of the 645s work well with a waist level finder. Two other great cameras but more on the extremes are the Fuji G/GX 617 or the Fuji 680, love both of them, still own a 680
try as you might, you shall not vanquish my unrealistic enthusiasm! *strikes heroic pose in towel cape*
what have you done?!
now i NEED that crazy flash umbrella satellite dish thing... AND a hat like a 1920s reporter!!!!
Then you want a Speed Graphic, not a TLR, it's a reporter thing -- And those men had to be body builders or something, because my Pacemaker is HEAVY and its a baby compared to the bigger ones.
Bob is right though, you will get weirdness with a ringlight that isn't ringing the lens properly. That doesn't mean you need to change cameras if you like the novelty of a TLR, although he has listed some really good ones. People rave about the Mamiya's and the Hasselblads, that Fuji 680 he mentioned looks sweet too. And closer to my price range
What you will probably need to do is get a flash bracket and use a different flash if you are serious about shooting indoors. I think most people who play with them now dally around outside with them, kinda like walking the dog, a getting some fresh air thing.
Anyway.. you should just do what you want to do. You've gotten a ton of suggestions, plenty of options out there but you need to deciiideee. Good luck!