I was connected to Copsewood Ladies team a few years back, who made it up through the leagues and made it up to the same level as the CCFC ladies team, and even beat them on a couple of occasions! Looking from the outside it seemed to me that they were poorly run as they should have been light years ahead of Copsewood, who were basically self funding through player subs and had to call on a number of players either disaffected by or deemed as not good enough by CCLFC.
CCLFC meanwhile had close connections with people on the Sky Blue Trust who used their influence to help with financial support, while they also had the pick of all the best age groups in the City feeding into them (ironically I think they also led them under the Coventry United banner). They would also get publicity through the Telegraph and CWR which was never afforded Copsewood even though they were in the same league.
On the point of whether CCFC should support the ladies team I would have to say it's a no from me. It would simply be an uneccessary drain on resources. The FA have done their utmost to turn the higher echelons of ladies football into a replica of the mens game, which I think is a shame, leading to Manchester United forming a ladies team and immediately being parachuted into the Premier League. Its funny looking back on those Copsewood days as they also were in the same league as the likes of West Ham, Spurs and Leicester, who they also beat on occasions, but are now all part of the ladies Premier League set up. On the flipside teams like Doncaster Belles, and to a lesser degree Copsewood are deemed to be not worthy and FA obstacles are put in place effectively forcing them to be unable to compete and disband.