Pausing between the minutes I play is just as important as playing. I do feel how my hands and arms tire and that effects the sound a lot. If I play too long I learn how to play false notes because I can't even hit the right places with bow and fingers properly anymore.

Ferenc Simon said
The way you describe it makes me think more and more that it's a simple matter of 'hidden tension'... some of your muscles are tense and you just don't notice it, which makes your hand tire waaaaay before it's supposed to. 

Try experimenting by putting down the bow and use your bow hand to reach over under the violin and go through your entire arm from shoulder to hand trying to feel for tense muscles, then if you find any, while you are pinching it with your hand.. try to consciously relax that muscle while still being able to work on the fingerboard with your left hand.

You can also check if your violin maybe has too high of a string action and requires more force than usual to press down the strings... Normally playing the violin should almost be effortless, you only need to put a slight weight in your fingers when pressing down. Same goes for the bow, normally when playing these simple tunes the weight of the bow is enough to make a good sound, rarely do you need to put more weight on it

Other than that it's not so much about style... I get it that you don't like the sound of the 4th finger and that the music you want to play doesn't really use those anyway, but it's more about developing yourself for YOU and not for your teacher or anybody else... since all these different techniques and styles are ultimately going to make you better at playing the style that you like so if I were you I would still try to experiment with developing that 4th finger as well.. who knows.. you might some day need it   For example when I went to a teacher a few years ago for some singing lessons, he was a classical teacher and most of the stuff we did was classical pieces, which I very much disliked at the time.. was even frustrated with them... but despite that I tried to do my best... Guess what.. the result was a major improvement in all the OTHER different styles that I prefer even if the techniques are totally different.. the muscle control I learnt there goes a long way... plus as a bonus I can now use all the sheet music of the stuff we used to sing to practice on the violin haha. so like I said.. you never know how useful it could actually be... that's why it's important to focus on the 'developing yourself' part.  

That sounds profound what you write there. Interesting that you also took vocal lessons, I also had a voice teacher in the late 80s who was an opera singer. Having learned classical opera singing, it's very hard to get rid of the vibrato. And I have to admit, it's easier to sing with vibrato if you have a bad day and a frog in your throat. Plus it's easier to sing with a 'covered' voice and then a little more forceful. Whereas the open and very lyrically light baroque style is very risky because you hear any slip of the voice, if the frog moves a little in the throat. (I'm not sure whether the expression 'covered' is correct, in German it is "gedeckt" like covered organ pipes are "gedackt" which is ancient German). Alto singers like Brigitte Fassbander and Kathleen Ferrier used to sing with a covered, very dark sounding voice and lots of vibrato. And I find that very interesting :  especially the 4th finger D and G have a similar quality. It obviously matches Brahms played with a strong vibrato. Whereas the open string alternatives  sound similar to a baroque singing voice. I tended to alto when I was taught by that opera singer and developed powerful high notes. But I can sing higher today after having retrained myself to a lyrical, open baroque mezzo. In the 1600s I can absolutely pass as soprano.

I am mainly an improviser at the violin. So I can adjust my speed to my form of the day. There a days when I get out of bed and can't go through a door without bumping somewhere. On days like that I lack so much power that my fingerboard arm tires soon. But it's possible I have hidden tension as well. At least I don't have to play THAT SCALE or whatever. On good days I do trickier things than on bad days. I did the same as jazz trumpeter and people at jam sessions said repeatedly I played like a professional. But since the late 90s I love baroque more and now it's especially 1600s violin sonatas, which are influencing my improvisation style.

I had my teacher for 7 months and she was brutal at times. She was pressing on the back of my hand, tearing my thumb, she downright made me angry but I held back. She wanted me to go one certain way, it  was the way she was taught at the music college. I do now profit from the basics the taught me. But I'm optimistic because I never force myself. There are people who have lots of vital energy, others lack that energy. I think we're just born that way. On trumpet I have learned to household with my energy too—especially on high-notes, using rather little pressure of breath.

But what you said sounds very profound! I'll watch myself whether I might sense tension in a certain place. It is mainly the 3rd finger. The 4th finger is not completely jobless of course, doing the black piano keys. Yes, I see that the little man gets stronger and those half notes fully occupy him.