Update: So I've been doing some more reading online and I might have found the problem, or perhaps part of the problem. After doing some adjustments, I'm receiving RF right now. (But I have before too so I'll let it run for some time and see if it stops receiving RF, or if it carries on now.)
Here is what I did. Maybe someone more knowledgeable on this stuff can tell me if this might be the fix, or if what I did has nothing to do with anything. :)
I realized the PK-232 manual says the box by default looks for 7 bit and even parity. I lost track, after so many adjustments and trying different things, of whether I had the ports setup the same. So I decided to give the PK-232 a "rest" command, then I exited the terminal. I set the COM5 port settings to 9600 baud, 8 bit, no parity, and no flow control. Then, I set Putty to the same specs and opened another terminal, and then did an auto-baud routine using the "*" command.
After the PK-232 came up, I set the following commands:
I then exited Putty, brought up APRSIS32, and confirmed the port settings for the PK-232 were 8 bit, no parity, etc.
So far, RF coming through. (I have internet disabled just to see cleanly.) My uneducated theory is, if this does fix the issue, then maybe I've wrongly convinced myself that the problem was not with the PK-232 settings and that it was "working" with tx before but not recognizing the AX25 packets because the port settings were a bit out of sync and it wasn't handling the last bit properly or something? (And I would have to also then assume that the similar problem I have been having with Mobilinkd is related to a power saving setting on the Bluetooth port, I suppose.)
I'll enable internet for normal operation and give a SITREP after a few hours, but I wanted to share the above for the good of the order, either (a) so someone can say the above isn't the issue, or (b) if it does happen to fix the problem, the next sucker who tries this can maybe save a few days of headache!