Quick update on my search, after having done a bit more homework:
VFD drivers seem to come with a significant output voltage drop as well, it seems. E.g. the MAX6921 outputs stay at least 2V below the supply voltage. Not critical for driving VFDs, but unfortunately a problem for me. Too bad, as the form factor would be nice, and SPI control should work for my application.
Low-side drivers for MOSFETs seem like the most promising route. The IRS4428 may not be the ideal candidate, since it also has a 1.4V output drop, and suspiciously its current consumption is not specified. But the NCP81071C (ON Semi) is very similar, with a manageable supply current below 2 mA, and the 2EDN7524/7523 (Infineon) offers two either direct or inverting drivers, with approx. 1mA supply current. All of these are specified up to 20V, matching my needs nicely.
Time to order some samples, I guess... Thank you all for your input, and special thanks to Zero999 for the low-side driver idea!