Pokemon-go – What determines whether a gym shows its banner

I'm pretty sure a Pokemon needs to be at full HP to be placed in a gym, so the reason yours wasn't available was because it was hurt from the previous battle. Heal it to full with potions and it will be available.

On average, there is a slight disadvantage to picking the weaker team, but not one that will completely affect your game.

For most of the game, your team affiliation will either not affect you (spawn/catch rates). There used to be rumors about giving you a perk for that team, but there's been no real evidence since. (Even after the game was reversed engineer to see more of the "gears", like how many steps to make a spawn call.)

The disadvantage comes from the lack of opportunity to earn the Defender's Bonus that Gym Leaders get for having a Pokemon at Gym. If you join a team that is the weakest in your region, then you and your team will have issues in holding onto Gyms. (whereas the teams with more members can statistically attack, take over, and fortify easier.)

The biggest takeaway though is that this is only an opportunity cost. The weaker team might be stronger a mile away, the stronger team might not frequent the area you are as much. The number of variables are unlimited as to that the weakest team might control all the Gyms of a city since their members are organized. There's no hard-set disadvantage, just a fluid difficulty curve.