If your water naturally has ammonia in it, then you don't need the fish to cycle it. I would put the fish in an alternate container, such as a rubbermaid bin, or other temporary container and use R/O water with a remineralizer until your aquarium has cycled using the ammonia in your tap water. It will not cost you $100 a month unless all the pet stores around you are from another planet, in another galaxy, where RO/DI units are fueled by diamonds, hahaha.
At most pet stores, the way it works is they sell you a 5 gallon jug and charge you $15 for it the first time. You don't have to use their jugs, either, you could get one off of craigslist or freecycle. Every refill after that is about $0.75. So if you had a 3G rubbermaid bin you kept your fish in and did a 100% water change every 4 days for a month, the water itself would be less than $10. You would have to get a remineralizer like Seachem Equilibrium, which is about $8 at a pet store. The most money you will have to spend is around $30, if you buy the jug, and the remineralizer, plus a month's worth of water. The next month after that would be less than $10.
I think you really need to investigate more alternatives before you euthanize the poor guy. If you live in an area that has some clubs or aquarium societies, you could probably get RO water for free from a saltwater or discus keeper. They usually have their own RO/DI units at home, and in my experience aquarium society people are really super nice.