Add noise/signal to mel log spectrogram
Except Logarithmic operation, other operations could be done on combined signal or separate signals both will give same result. 


$STFT$($n$) $+$ $STFT$($S$) $=$ $STFT$($n$ $+$ $S$)
$mel$$(STFT$($n$) $+$ $STFT$($S$)) = $mel$($STFT$($n$ $+$ $S$)))


so just removing effect of logarithm and adding should do. 
though I am not sure that artifacts will go or not, but you need not do reverse Mel(I am assuming that mel filter banks central frequencies and bandwidths are same for both $S$ and $n$).