Birding is a “natural” extension of photography for many. After all, I am willing to bet some of every photographers first shots are ducks
But joking aside, so you can make a good decision.. Pentax AF works reliably with optics that have a combined / native aperture of F6.9 or less. A native aperture of F8 only works on the very best of days. This goes as far back as 1990 and the PZ1.
I base this on personal experience using F4.0 lenses like the K300/4 with the 1.7x AF converter. That works very well even in shaded conditions, and also stacking my sigma 1.4x behind the 2x on my 70-200/2.8. On a sunny day although it would hunt quite a bit it could achieve focus lock.
Therefore a 1.4x behind the 300/4.5 will work
As far as image quality, the issues with a TC are that they magnify everything, including all the faults of the prime lens. If you have lateral CA, a TC will give you more, but a good TC does not really degrade the image significantly, from that of the lens it is attached to.
As some have noted, one big advantage of a TC is it does not modify close focus, and your 300 with a TC might be better than some longer lenses if you are in close. What this means is don’t get rid of your 300 and TC if eventually get a longer lens. It still has some uses.
The other advantage is the TC on your 300/4.5 will be a lot lighter, and again even if you get something bigger and badder for your main birding activities, you might want a light option for travel. I would never (probably) take my DA560 on an overseas trip, but would take in its place my trusty old Sigma 70-200/2.7 and 2x TC. Lighter, smaller, and much more versatile overall, when considering true birding activities might only be 10% of my trips time.
Even when you get a TC, and don’t worry about any vignetting you may get even with the Pentax 1.4x which is only for crop sensors. If you are close enough to a bird to be using the full frame of a K1, you don’t need the TC any way. Most of the time you will still be cropping in. From a lot of experience I can tell you that it is very rare that with my K1 that I do not crop even with the DA560. The sad reality is that for birding, you NEVER have a long enough or FAST enough lens. You simply make due with what you have.