Last time I created a manual entry, I didn't get the APPLY button until I
entered a Gateway. I needed one - not sure what you can put in that field if
you don't.

Well, 192.168.5.299 isn't a valid IP address. Was that a typo in your post?

it was a fat-fingered mistake. I meant 192.168.5.200.

As soon as I change anything in the Network Manager Wired Connection dialog, the
APPLY button is grey'ed out and it will not come back regardless of how many
fields I populate.

Have you tried manually setting up your card?
It is done by writing: sudo gedit /etc/network/interfaces in the terminal
There you will probably find this written:
auto lo
iface lo inet loopback

Type the following info:
auto eth0 (your eth may be eth1 or something else)
iface eth0 inet static
address - here you put your IP number - eg. 192.168.0.200
netmask - your subnet number - eg. 255.0.0.0
gateway - your gateway number - 192.168.0.1
broadcast - based on IP - eg. 192.168.0.255

Save and exit
Then type:
sudo gedit /etc/resolv.conf - note how resolv is spelt. This is not a
typo.

The file may be blank. It should contain your DNS addresses. To add them
manually - type:

nameserver DNS address
nameserver second DNS address

save and exit.
Your card should be manually set up. Then type
sudo /etc/init.d/networking restart
to start your network

I tried that (along with setting up a
/etc/rc.conf file) but it did not work and it interfered my wireless connection.
I am confused as to what to use for DNS addresses as in my final configuration
this will be a hardwired connection between my laptop and the application card,
with no router or internet connections anywhere.

If your connection will be between two devices on the same hard-wired network,
and if the wired connection is going to be used *only* to communicate between
those two devices, then you don't need DNS and can make all your connections to
IP addresses. DNS is only required if you need to connect to a device/site that
you only have a name for.