Are you using a water conditioner to dechlorinate the water? I also notice there are a lot of bubbles on the sides of the bowl. This means the water has not been aged. You need to let the new water sit for a day or two in a bucket to "gas out".* Water that has not been aged can cause something called "gas bubble disease" which can be fatal.
What is his water temp? Bettas need warmer water of about 80F. Cooler water makes them lethargic and constipated.
Also, if you've been feeding him "as much as it will eat in several minutes 2 times a day"* then he is definately constipated! Bettas should NEVER be allowed to eat as much as they want. Their stomachs are the size of one of their eyes. Pellets expand to about 3 times their dry size after being swallowed. He should be fasted for a few days and if he will eat, give him some green pea, (frozen pea, microwaved or defrosted with skin removed). He should only have a VERY small portion of pea, about the size of one pellet. Give maybe two pieces that size twice a day. Bettas can survive a few weeks without food so don't worry that he will starve in a few days. I give my Bettas one pellet per meal 4-5 times a day for the males and 3 times a day for females.
I'm afraid without a larger heated tank constipation is very difficult to cure. Cold water will slow his metabolism and keep him sick. Those "Betta Kits" sold by the petstores are really Betta Killers. I've never known anyone who could keep a fish alive very long in those. It is too small for a filter or heater which means the fish is too cold and exposed to high levels of
ammonia in the water. The water would need to be changed every day to prevent ammonia poisoning. This stresses the fish as well. You may get away with changing the water every few days if you use a water conditioner called AmQuel+ or Prime, both of which neutralize ammonia, nitrites and nitrates. You need a
test kit for at least ammonia (the liquid dropper test not the strips which are not accurate) to keep an eye on the ammonia. It should be 0. If you read ANY ammonia it is too much.
Try the pea treat and age his water and keep his water clean and ammonia free. Keep him as warm as possible. Get a larger tank (at least 3+ gallons) with filter and heater asap and make sure it is cycled (people here can help you learn how to cycle a tank). Hopefully he will make it that long. Sadly though, helping him in this tiny cold box will be very difficult. I wish these Betta Kits were banned from stores along with small bowls, lava lamps and all the other ridiculous things people sell as "Betta tanks". I'm sure you bought Dodger with the best of intentions but were seriously misled by the petstore about what is a suitable home for a Betta. Bettas are cheap to buy but are NOT cheap to care for. I've spend probably $2000 in the past 7-8 months on products, tanks and medication for my Bettas.
Good luck with Dodger and keep us updated. I hope he makes it and that you can get him a proper tank asap. Please feel free to ask as many questions as needed. Read through the posts here. You will find many of us have similar experiences that may be helpful to you.
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I will let the experienced people hand out the good advice about your fish, since I have only had mine for about 2 months. 
