Before anything, I don’t believe that it is possible to have one Wi-Fi router—or perhaps one unique SSID—set to use mixed WPA2 and WPA connections. That said, I did discover some hope on the Palm TX WPA2 capability side of things. Read on.
So you say this; bold emphasis is mine:
There is no way to upgrade the Palm TX, so any connection would need to be accomplished by making some kind of exception in the router
to allow connection of this one device using WPA. Is there any way to
While the rest this answer does not technically answer the core question of setting a WPA exception for one device, based on this March 2010 discussion in this official Palm support thread, a firmware update for the Palm TX that provides WPA2 connectivity—among other things—is available:
You can purchase an upgrade for the Wifi software that will enable
While the URL provided seems dead right now, the Internet Archive’s Wayback Machine is quite helpful in providing an archive of that page. That page has links to these documents:
Now with all that said, it seems that while the update was once a paid update in 2010 it’s apparently free as of 2013? Or at least someone at HP—which owns Palm assets nowadays—has no problem with a power user on their forum publicly sharing that update for free as of October 2013?
The Dropbox URL a volunteer “HP Expert” provides grabs a file named,
TX Security Upgrade.zip which seems to be the real deal. I assume that since the Palm TX is a discontinued/vintage product nowadays HP/Palm has no problem with others passing it along that firmware for free nowadays.