well first, you should be looking into an upgrade. 1gallon it too small. bettas are tropical fish and for optimum health they should have at least a 2.5gallon tank with a small filter and a heater. without this the fish's health suffers, as you have noticed. in tank without filters frequent full water changes are needed, but also very stressing. unless you upgrade it's likely you'll run into this problem again... you'll also be shaving years off your betta's life

anyway, cutting up the tablet is perfectly fine. just try to get it as even as possible.