No... even with water conditioner your spring water is far lacking what is needed. Tap water, whether you believe it's gross or not, is healthier and safer to use over spring, filtered, distilled water.
Slowly acclimate him to tap water by doing partial water changes over a few days time with tap water in the tank. Spring water is not sufficient, and conditioner removes and binds elements in the water, but won't give the water what is needed. Nutrition comes from food, minerals is what the spring water lacks and is needed by the fish.. people want to believe what is good for us, is good for the fish.. such as spring water.. but they are fish and their needs are different.
Myates is right, tap water is much better. Distilled water is pure water, no minerals. Think in the wild, do you really think it is only water? Nope there are a bunch of things in the tap water that will keep your fish healthier than if it was in bottled.
And baby bettas are harder to care for and won't easily forgive mistakes. Adult fish are much harder than babies, so extra care should be considered.
Spring water is good... Usually.. Test the ph. Sometimes spring was has ph inadequate for fish. I think people are confusing spring water with distilled which removes everything. If your "spring water" is actually distilled or reverse osmosis (r/o) water then use seachems equilibrium to add back essential trace minerals
I would not use the water you have from the well with your fish
Spring water, from a bottle or a jug you buy at the store is what we are talking about, and the filtration process removes certain minerals that the fish need. Well water tends to be just fine and preferred by a lot of people keeping fish.
The slime part, is it a film on top of the water? That is usually protein and common, regardless of where you get the water- spring water lacks, so you may not get it in that. You use a paper towel and scoop out the protein when it builds up.
Seachem Equilibrium adds in minerals for a naturally planted tank. What is recommended to people who don't want to use tap/well water is half and half- so at least there are some minerals in the water for the fish to stay healthy.