Hi Martx-5,
I don't have a TR3 (yet) but am fairly certain 6" wheels would be a problem on a stock TR3, particularly with wire wheels that will give a slightly wider track once the adapters are fitted to the car. 5.5" might work, but even those could be a too wide on certain cars. I have 5.5" alloys on my TR4 and those leave little room to spare.
There is some difference from one TR3 to the next, since the cars were essentially hand built. So, wider wheels might fit on one car, but not the next. So, unforunately, the only way you can tell for certain is to test fit a wheel and tire combination and see if it has clearance. A local tire & wheel store might work with you, but a catalog/mail order place likely won't allow returns on wheels after tires have been fitted. Maybe there are some other Triumph owners in your area, who have various tires and wheel combinations and would be willing to let you test fit one on your car?
Also, I believe a 6" wheel is a bit of overkill for a 185 tire. I'd think 5" wheels might be adequate, and are more likely to fit the car okay, if you can find a set of those instead. (Tire manufacturer websites often provide wheel width requirements).
Let us know what you find out.