If you convert the .srt files to .ass, then you can have a lot more control over how the subtitles appear.
To convert use ffmpeg:
ffmpeg -i input.srt output.ass
This will give you an .ass subtitle file. You'll see in the header of that file that there will be details about font, size, placement colour and so on. Here's an example from Wikipedia:
Format: Name, Fontname, Fontsize, PrimaryColour, SecondaryColour, OutlineColour, BackColour, Bold, Italic, Underline, StrikeOut, ScaleX, ScaleY, Spacing, Angle, BorderStyle, Outline, Shadow, Alignment, MarginL, MarginR, MarginV, Encoding
The first line lists all the settings, and the second lists the values for each style.
You can edit the header by hand to get the size and placement you want, and then burn it in using ffmpeg again:
ffmpeg -i input.mov -vf ass="output.ass" output.mp4
You may want to add encoding settings etc, but that's basically it.