100% water changes are important because your fish is constantly excreting ammonia from his gills as he breathes. This ammonia is constantly building up in a small quantity of water since you have no bacteria to break it down into less harmful substances, and no plants to consume it. If you change 50% of the water today and 50% of the water tomorrow, technically, this adds up to 100, but that does not mean 100% of the water has been changed. Inevitably, some of the old fouled water is left over plus whatever the fish has added in the meantime. Over time, this method becomes less and less effective and the fish is never allowed to live in the pristinely clean conditions that are ideal. Just a lower degree of disgusting deathsoup.
Changing 100% of the water is not as distressing as being constantly exposed to even small amounts of ammonia. If you do it properly and take the time to acclimate your fish to the new water every time, the stress caused by it will be minimal, and the fish will greatly benefit from clean conditions.
Honestly, you are going to continue to have issues with that tiny tank and the sooner you come to terms with getting rid of it and purchasing something more practical for everyday use the better off you and your fish will be. It doesn't have to be anything fancy, you can use a rubbermaid or sterilite plastic storage bin until you can afford something that looks a little nicer. You may have one already or you can borrow one from a friend. These bins are only about $3 for a 2-4 gallon size bin, and are safe to heat, easy to clean, and very durable.
You can use the money you save on the container to purchase a proper adjustable heater, like this one: http://www.fosterandsmithaquatics.co...8&pcatid=11368
heaters are not optional in betta keeping, and the light you're using to keep him warm is likely adding to his stress level. Bettas come from an area of the world where the water is dyed a dark tea color by dead leaves falling into it, so they can be stressed out by bright lighting. They also need a day/night light cycle, so whenever you turn the light on and off you may be subjecting him to dangerous temperature fluctuations.
Frozen food is not a bad idea. I keep mine in a brown paper lunchbag, but even if I didn't, they just look like opaque ice cubes and nothing gross.