I don't imagine many people will be reading this, but I wanted to follow up in case anyone else comes across this same problem down the line.
In summary, it now looks to me like this has something to do with my Linux environment's vector-writing software. The script below produces a series of figure files in different formats (of which pngconvert just uses ghostscript to do the EPS to PNG conversion) ...
x = 1:1:10;
y = x.^2;
figure(1);
plot (x,y,'b-', x, y, 'ro');
xlabel ('Property 1 [blah{\circ} blah^{-3}]');
ylabel ('Property 2 [blah° blah-3]');
exportgraphics(gca, 'test_PDF_vec.pdf', 'ContentType', 'vector');
exportgraphics(gca, 'test_PDF_img.pdf', 'ContentType', 'image', 'Resolution', 300);
exportgraphics(gca, 'test_PNG.png', 'Resolution', 300);
print ('-depsc', '-painters', 'test_EPS');
pngconvert('test_EPS.eps', 300);
Of these figures (and I've attached all bar the EPS one; this environment doesn't like that), the vector ones (including the PNG created from the EPS) exhibit the space problems in the units, regardless of whether they're printed using the print command or exportgraphics. Meanwhile the image ones (including the PDF) do not have this problem.
I'm not familiar enough with my Linux environment to dig into this [*], so I don't have an answer for why this happens, but as I can now produce image files in the way that I want, I'm now happy.
[*] I've loaded texlive/2018 to access ghostscript if that helps!