It would be useful to speak to the dentist that has supplied you with the brace. Some can be kept in as they have a special clip that secures them stops the chance of the brace dropping in to the throat and causing choking. It would also be required that a gum shield was worn as it protects the back of the lips from the fittings on the brace.
However, it is down to those that monitor and govern your Rugby as there is a health and safety issue that they must satisfy themselves on before each case is allowed to play.
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Under IRB Regulations you would be required to remove the facial adornment by a referee before you were allowed to take the field. Wearing this type of thing is HIGHLY dangerous and can cause significant injury.
Even the wearing of wedding rings is forbidden
No piecing is allowed on the field of play and any person wearing a ring either must remove it or have it completely and soundly taped over - these are required under safety rules