You can try activating the autosave feature if you're good at typing. This 
means that the translation will be saved as soon as you click in the next 
textbox. 

I use it like that, and used to have it as default, but most users asked for 
the save button.

I type the translation, press tab to move to the next translation, and the 
previous translation is saved automatically in background.

Refresh after saving is something to be used only on the review step.

So now there are two options, can you think of another one ?

Option (3) would be to save *every* filled text box text when I hit any save 
button on the page.

The problem I have with saving is that it lasts a long time, and interrupts 
work.  If I have to press "Save" every time, and then refocus with mouse, I am 
interrupting work waaay too much.

On the otherhand, the 'autosave' option in Narro is too erratic to be useful. 
More often than not, the auto-save interrupts me in mid-typing and it saves a 
partial translation that I then have to mouse back into, and correct.  I would 
simply remove the autosave, because for practical purposes it does not work. 
(That is, it doesn't work robustly, but for high-volume translation that's 
exactly the same.)

The current work pattern is:
<translate>
<save>
<wait>
<translate>
<save>
<wait>
<translate>
<save>
<wait>

If you allowed multi-save, it would be something like:

<translate>
<translate>
<translate>
<save>
<wait>
<translate>
<translate>
<translate>
<save>
<wait>
...

which should be way faster than the old way.

With the save button, I'd do it like this:

translate
press tab to focus on save, press enter
press tab to move to the next text
translate

You don't have to wait, saving is done in background.

Multi saving is costly (you have to wait), but I could have the button there, 
top and bottom as it was in the past.

Also, there are several checks done, so you might need to fix a few or ignore 
the warnings on pressing save all.