Low water temps can cause lethargy, lack of eating and slow metabolism - fry do best and is recommended to be in temps of 82 - 84* - lower temps would cause them some health issues such as lowered immunity. Recommended to raise the fry tank temp, and to get him his own heater and set it to 78-80+ depending upon how old he is.
With snails in that small of a tank you definitely want to be doing plenty of water changes as they produce a great amount of ammonia. If your tank is filtered then you will want to be doing 50% water change twice a week - with siphoning of the substrate once a week. That is the recommended for a normal tank, and a tank with a very young fish you definitely want to be doing that, possibly even adding in another 25 - 50% per week until he is about 4-5+ months old to prevent the hormone that stunts their growth from building up. Being so young they need really clean water.
With a tank with NO filter you will need to be doing 1 50% and 1 100% water change per week - that is a need for tanks that size with no filter. Add in another 50% per week since he is very young and since you have snails.
My personal suggestion would be to remove the snails, do 2 50% and 1 100% water changes per week with water conditioner, and raising the temp to 80 - 82* in his home tank. Young bettas need extra care to be healthy... it sounds as if improper temperature and water changes are causing him to decline in health quickly. Young ones aren't as hardy as adults.
Don't worry about changing the pH drastically, as since it is from the same water source it shouldn't be that big of a difference. If you are worried about that then you can remove him and place him in a cup, change the water, then float him in the tank removing about 25% of the water from the cup and replace it with water from the tank - do that 4 times about 5-10 mins apart and he will then adjust to the new chemistry. But normally when doing 50% water changes, you don't have to worry about the pH difference.