For King Dede....I can't tell by the pic what the lump is, however, I can help you with the long term care.
The 1gal unfiltered is fine to use for long term keeping provided that you maintain the water quality. Since you don't have a heater-do you have a thermometer to check the water temp-Even if you have to use a thermometer from the kitchen to check the temp-you can use a plastic baggie over the tip and stick it in the tank to get the water temp. Check the temp and post that
Then do a 50% water only change leaving the Betta in the tank-just dip half the water out and replace with dechlorinated water that is close to the temp of the tank-do this for the next 2 days-then get on a regular twice weekly water changes of 50% water only and 100% every week.

OFL, King Dedede (my first betta) had a lump yesterday, it was near his gills, and now his body is turning black with some white on it. He's also lethargic. Do you know what's happening to my betta?