First the battery tender plug. The tender is a light guage wire, and only good for about 5 amps. So, no, do not use the tender plug in. You can switch to the Gerbing battery lead, which is a heavier wire, and has a 15 amp fuse. Then you can use that plug for either. Or, you can stack the plug wires on the battery and use each, but that is messy.
The second is the capacity of the electrical system in the wing. It is about the same as a full sized American Pickup, so it will run just about anything, except maybe an arc welder. So, no problem with the output being able to charge the battery.
Third thing is when to plug in the suit. Plug it in anytime. I just wait until after I start the bike, just because a starter draws a lot of current, and it just lightens the load on the battery while cranking it up. But either way will not hurt a thing.