Killing the battery I highly doubt it, wicht is the original question.
Battery are rated per discharge cycle, running it low a couple time a year I highly doubt will be damaging.Do not leave it discgarged do. If you have a way to mesure the amp going in and out will charging, it would help to let you know to stop charging, when charging goes under 10amp lets say, you can stop the geny it's not efficient anymore. Make sure to charge good when your back home.
I would recommend to put your your 10 amps charger if it a 3 stage (it should keep your battery higher then 14v longer then would the wfco).
Upgrading wire to and from battery to converter is not a bad idea.
Save your money, you got all you need.
Here his a cheap battery monitor that could help manage the needs...
https://www.amazon.com/bayite-6-5-100V-Display-Multimeter-Voltmeter/dp/B013PKYILS/ref=sr_1_3?ie=UTF8&qid=1518876299&sr=8-3&keywords=12v+battery+monitor