I do rather like the word “princess”. Its dignity and grace seem to have a life of their own, which no number of waspish and balding royal spinsters can quite suppress. I suspect that the grating quality of words like “actress”, “doctress”, “stewardess” owes more to the fact that they are artificial coinages, made necessary by the fact that women only recently began to have these rôles. The fault, if there is one, then lies more with society than language. Old gendered words, such as princess, countess, redemptrix, châtelaine, vicereine, queen, canoness, poetess etc. feel beautiful, natural and strong to me - they each contain something that is not quite contained in the sense of their masculine counterparts; but they may strike you differently.
Reading your post, I was reminded of a passage in one of my best-beloved novels, C.S. Lewis’ Perelandra; he was, of course, a very erudite linguist himself, much like Tolkien. It’s in the spoiler below.
Both the bodies were naked, and both were free from any sexual characteristics, either primary or secondary. That, one would have expected. But whence came this curious difference between them? He found that he could point to no single feature wherein the difference resided, yet it was impossible to ignore. One could try–Ransom has tried a hundred times–to put it into words. He has said that Malacandra was like rhythm and Perelandra like melody. He has said that Malacandra affected him like a quantitative, Perelandra like an accentual, metre. He thinks that the first held in his hand something like a spear, but the hands of the other were open, with the palms towards him. But I don’t know that any of these attempts has helped me much. At all events what Ransom saw at that moment was the real meaning of gender. Everyone must sometimes have wondered why in nearly all tongues certain inanimate objects are masculine and others feminine. What is masculine about a mountain or feminine about certain trees? Ransom has cured me of believing that this is a purely morphological phenomenon, depending on the form of the word. Still less is gender an imaginative extension of sex. Our ancestors did not make mountains masculine because they projected male characteristics into them. The real process is the reverse. Gender is a reality, and a more fundamental reality than sex. Sex is, in fact, merely the adaptation to organic life of a fundamental polarity which divides all created beings. Female sex is simply one of the things that have feminine gender; there are many others, and Masculine and Feminine meet us on planes of reality where male and female would be simply meaningless. Masculine is not attenuated male, nor feminine attenuated female. On the contrary, the male and female of organic creatures are rather blurred reflections of masculine and feminine. Their reproductive functions, their differences in strength and size, party exhibit, but partly also confuse and misrepresent, the real polarity. All this Ransom saw, as it were, with his own eyes. The two white creatures were sexless. But he of Malacandra was masculine (not male); she of Perelandra was feminine (not female).