Are you on well water? I see you don't use a dechlorinator
Your temp is a bit low-I would recommend a heater and keeping the water temp in the 76-80F range. Some Bettas are more sensitive to water temp than others...adding a heater may be what is needed.
I would increase the water changes-10g without filtration or live active growing plants needs at least twice weekly 50% water changes and a 100% every month or so to maintain water quality
I would add the filter and get the tank cycled-you can add a baffle to the filter if it seems to strong of a current for him (look on this site for a DIY baffle information) once the tank is cycled making 50% with vacuum weekly should keep water quality good-provided that you don't over feed and uneaten food is removed after feeding him.
Adding live plants can also help.
You may have posted this and I just missed it...but....how is his appetite? and any other symptoms like flashing or rubbing himself on objects in the tank and can you see anything on his body that looks like grains of salt or a gold powder look?
At this point I would recommend that-add the filter to the tank and cycle, get a heater for the 10g
For the Betta-QT and start making 100% daily water changes for 3 days and see if he will perk up, if not, add aquarium salt 1tsp/gal along with 100% daily water changes for 10 days
If he perks up after the 3 days of fresh clean dechlorinated 100% water changes-add him back to the 10g after you make a 50% and vacuum-acclimate him properly before adding
Make sure water temp is with a couple of degree of old and new water with water changes so not to cause temp related problems and use dechlorinator if in city water supply.
Or, if you can't Qt-start making daily 50% water changes on the 10g for 4 days and then twice weekly unless you add the filter then 50% weekly with vacuum and add a heater
Even when the water looks clear-it can still contain ammonia, nitrite, high nitrate and DOC's and the only way the fix this is with a water change
Anytime your fish is acting odd or change in behavior-make a 50% water only change to rule out the environment as the cause