you will need to place some of the gravel from the 10 gallon aquarium and place a small plastic plant there a very very very weak air pump would work fine, get a plastic tube that spilts the airline hose and place one in your main tank and one in your betta tank, tie knots on the correct one to get a measured amount of flow in each tank, the betta tank should have very little air, maybe like a bubble per 10 seconds or something, this will keep the tank a bit more oxygenated which will help speed up the healing process. Also when you go to get medication you can try buying some "betta water" it is just normal betta water that has been highly oxygenated for the betta.
make half of the tank this and half of the tank normal tap water
i could be wrong but i believe i read on a site that aquarium salt will not help.
but it stated that epson salt will help draw out the fluids witch is causing the protruded scales..(pine cone affect)..most cases of dropsy is due to the kidneys shuting down witch disables the fish ability to expel necessary body fluids witch build up in there body causing the swelling witch results in the scales sticking out...from my research it is 99.9 % fatal and if it is caught very very early its treatable but most times whan the scales are protruding its to late..sorry to say..another sad fact is fish can live sometimes for weeks
or possibly months in this condition.