I don't think either of the choices look bad. However, finding a 4x4 tile would be most ideal, even though you still have to cut it down. If you are unable to find it, the 3x6 tile blends in with your current tile well enough. The bullnose draws the eye a bit too much, in my opinion.

i wonder whether the new tile will actually blend when its finished. the individual tiles look ok, but they could well match each other better than they blend with the exisiting, and it will look like a different color stripe along the bottom.

i'd search your wallet and consider replacing the entire backsplash at this time, rather than spending 'not nothing' and have it be eyesore-ish.

another option would be a different, more contrasting, band of tile along the bottom. not the bullnose, but not trying to match the field tile, either.

Since you have gone to the expense of getting new quartz counters why not spring for a less dated look for your backsplash it will be a much better choice than some band aid solution

Of those, I prefer the bullnose, which is always going to look more intentional, assuming you have a traditional kitchen. I think that the offset grout lines will seem odd, however well the new tile matches the color of the old. Any accent tile will draw the eye even more than the bullnose.

The counter top in the photos is the existing, correct? Which quartz did you select? Do you have a sample to show with the travertine?

I would not use the bullnose, I think the 4x4 or the 3x6 is a much better look.