One way to partially answer your last question might be the following. To each $f\in L^2(a,b)$, first associate its Lebesgue primitive $F(x)=\int_a ^x f(t)dt$, then define $Tf$ as one of the four Dini derivatives of $F$, e.g. $$Tf(x)=\limsup _{h\to 0^+}h^{-1}(F(x+h)-F(x)).$$ Then $Tf=Tg$ everywhere if $f=g$ almost everywhere, $Tf=f$ almost everywhere, and $Tf$ is continuous if $f$ is equivalent to a continuous function. Thus the map $T$ associates to all members of a class of equivalence in $L^2$ the same function, which is the continuous representative of the class when it exists. An additional advantage is that the method is 'constructive'.
One way to partially answer your last question might be the following. To each $f\in L^2(a,b)$, first associate its Lebesgue primitive $F(x)=\int_a ^x f(t)dt$, then define $Tf$ as one of the four Dini derivatives of $F$, e.g. $$Tf(x)=\limsup _{h\to 0^+}h^{-1}(F(x+h)-F(x)).$$ Then $Tf=Tg$ everywhere if $f=g$ almost everywhere, $Tf=f$ almost everywhere, and $Tf$ is continuous if $f$ is equivalent to a continuous function.