I have the 55-300 PLM and the FA*300 f4.5. I also have a Kenko 1.5x teleconverter. The TC will work with the PLM lens, but only in optimal conditions. Adding the TC is equivalent to a one-stop loss in aperture, so f5.6 (the maximum aperture on the PLM from about 136mm to 260mm) effectively becomes f8 and f6.3 (the maximum aperture from about 260mm to 300mm) effectively becomes f9. AF is very unreliable at these apertures. And as others have pointed out, those sorts of apertures often mean unacceptable compromises in shutter speed and ISO.
The FA*300 f4.5 works OK with the TC but even then the maximum effective aperture (f6.3) is not ideal for AF speed and accuracy. I expect the slightly faster aperture of the DA*300 f4 would make an appreciable difference, because Pentax cameras seem to focus a lot better with a maximum aperture of f5.6 than a maximum of f6.3.
The other thing is that (at least with a 24mp APS-C body) a TC doesn't always produce better results than cropping. There are some informative threads about this. Start with this one:
Understanding folks saying a TC doesn't ad more detail. - PentaxForums.com
The 55-300 PLM provides very good results, even at 300mm,
for a consumer zoom. Not only is the resolution very good, but the rendering is also quite pleasant. In fact I like the bokeh from the PLM much more than that for the FA*300. And of course the AF is very fast and quiet.
Have a look at this thread for samples:
HD Pentax-DA 55-300mm F4.5-6.3 ED PLM WR RE - Show us what it can do, what it CAN'T! - PentaxForums.com
You will notice that it's not just a wildlife lens - there are many fine landscape images in particular.
As for resolution, if the subject is close enough or large enough to fill most of the frame, and the light is good, the PLM can hold its own against the FA*300. For yard birds, for example, the PLM is a wonderful lens - affordable, light-weight, versatile, WR, nice rendering, very fast and quiet AF with QS,
But when resolution is tested by a subject further away, or where a wider aperture is needed, the zoom is no substitute for a 300mm prime.
(These three are all substantial crops.)
The other option to consider is a long zoom, like the Sigma 50-500 (Bigma) or DFA 150-450. These are much heavier (about 2kg) and, in the case of the DFA, more expensive, but they add more reach and the versatility of a zoom.