As a matter of rule, I checked the "New Rules of Golf for 2019" and there is no rule about which club to use where. In fact, the term "putter" or "wedge" is never used even once.
http://www.usga.org/content/usga/home-page/rules-hub/rules-modernization/golfs-new-rules.html
There may be a few local rules to consider for some clubs, and if you damage a green IRL you may get booted, but I don't see this as being illegal. It is a skill, and IMHO, you can't duff a putter, so why take the chance : )
PS - in real life, while putting, I use only my right hand, have so for 10 years or more. To practice, given the arm action and stroke needed, I sometimes use a wedge or even a 9-iron to putt edge on for a precision drill. I don't loft the ball at all but thought I'd share anyway.
Cheers!