In addition to the existing comments: I’ve checked with my ceramic expert (my daughter) and this is a summary of what she told me.
From a ceramic standpoint, a second firing should not be a problem unless you are firing at cone 10 or higher temperature, in which case the ceramic may crack in the second firing.
From a glaze standpoint, assuming the etched area has broken the glaze surface the new glaze should stick. This may still have to be hand painted as applying glaze to the entire surface then wiping the excess off of the non-etched area will likely wipe most of the glaze off of the etched area as well. Assuming you can get enough glaze applied to the etched area, here is where the problems really begin.
There will still be some of the original glaze under the new glaze and obviously around the area. The way the old and new glazes interact may in my daughters terminology lead to very wacky results. The glazes may just run together creating a mud puddle or the new glaze may not even stick to the old glaze that is under it.
Basically there are to many factors to make specific statements, but it never hurts to try and you may end up with something that works very well or at least produces something very interesting, not necessarily what you wanted.