Adjusting the vertical alignment line position in Premiere's Captions
I've found a workaround in Premiere, outlined below:

Make a copy of your .srt file, as you will need to import two separate files for this to work.

For example sub1.srt and sub2.srt

Import both .srt files into Premiere.

Place them one above the other on your timeline.

Make sure they start at the same timecode.

Set the text size and alignment as usual. Make sure you do this to all the captions by right-click on on a caption and clicking “Select all”.

Now modify the lower track, so that it shows a black background with black text only:  

Double click the subtitle track to select the correct captions in the Captions Tab (they all look the same, so this is easy to mess up).

Right click one of your captions and choose “Select all”.

Make all the backgrounds and all the text black

Now modify the upper subtitle track so it shows the text only with no background:

Double click the subtitle track to select the correct captions in the Captions Tab (they all look the same, so this is easy to mess up).

Right click one of your captions and choose “Select all”.

Make all the backgrounds have an Opacity of 0%.

Now select your lower track again, and reposition it in the Effects tab until the alignment looks correct.