The reason you need to do at least 1 weekly change on an unfiltered tank is because the water change is only diluting the ammonia.
If you take the waste out and do a 50% water change you still have the ammonia left in the other 50% of the water, so the next time you do a change you're going to have even more ammonia in your tank.
In my experience a 2.5 gallon tank with one betta should be fine with just a single 100% change a week, provided you aren't over feeding. That in conjunction with removing the uneaten food/waste with a turkey baster should be just fine. You can always top off if you have evaporation in the tank over the course of the week.
You can get your fish used to netting or cupping by conditioning them to the net or cup. First start by just feeding them with the net visible on the outside of the tank, then feeding them with the net inside the tank, then slowly work towards getting them to swim into the net for feeding. It can be done, just takes a little effort.