My experience is that people brush this kind of complication under the carpet, or at least it seems typically not mentioned in literature. The awkward problems of the year being 365 or 366 days -- and to the point here, it not being 364 = 7 * 52 days -- are important but not widely considered interesting!
If within-week variation were your only problem, then you could just look at day of the week as a predictor. Although you don't spell it out, I sense, however, that you also have seasonality and possibly trend, else why be interested in STL?
You don't spell it out either but I guess wildly at some economic data here. I also focus on Western calendar years.
I can't say I like any of the three solutions you suggest. What's more, I predict that if you chose any you would then get criticism from reviewers for arbitrary and indeed biased choices. (It's empirically important that the last two weeks of the year are special too, at least in societies that celebrate Christmas as a festival.)
I have not used STL and would tend to try to handle problems like this in a top-down manner. With daily data and days indexed 1 .. 365 or 366 I have used
fraction of year = (day of year - 0.5) / (365 or 366)
as a predictor. In any decent software you can identify leap years and non-leap years automatically, e.g. by 29 Feb being a valid date or not, or some day-of-year function being 366 or 365. I can tell you how to do that in Stata, but not in R.
Then day of the week is another predictor (e.g. as a set of indicator variables).
However, my experience is mostly with environmental data for which some sine and cosine terms often yield a fair approximation to annual cycles. That's typically not good enough if your data were economic.
There are some very elaborate seasonal adjustment programs intended for economic data that may do much more. I can't advise on details, but they tend to have special handles for holidays and unusual days, depending on what society they are constructed in.
