The battery voltage will drop rapidly. However, then it should reach a point where it will stay for a long time. What's the voltage that it's dropping to? How old is the battery? You should be able to get way more time than 4 10 minute flights.

Be careful with field chargers, some of them are setup to charge the higher capacity batteries used for electric cars and planes, so they may provide too much current for the typical transmitter battery pack. If it's specifically designed for charging transmitter batteries, it should be ok, but it would be good to know the charging current.