Question 120v generator hooked to a gridboss

I have a small 2400 watt generator that puts out a pure sine wave but only 120v, so only one leg of my 240v generator connection to the gridboss would be hot. Can this generator hooked to the gridboss still charge the batteries that are hooked to the flexboss21 inverter while it is supplying power to my house (no grid tie). During a power outage I would like to charge my batteries a little even if it's only a trickle charge.