I'm not familiar with the details of a TopFin filter, so I currently can not speak to how to keep one cycled. But I can show you a cheap filter that you can modify that will keep your tank cycled and go easy on your betta.
My instructions are going to include links to specific products. But all of these specific products are just examples. You can walk into just about any fish tank and find similar items you can use just as well.
1. Start with the Tetra Wisper 10i
Whisper In-Tank Power Aquarium Filters from Tetra - Filters - Fish - PetSmart
Throw away all the media that comes with the filter.
This pump pulls water from the bottom and it simply flows/spills out the wide open top, so there is no "jet" of water.
2. Buy some biopellets
Fluval® U2/U3/U4 Biomax Underwater Filter - Online Only Sale - Holiday Sales - PetSmart
These are going to be your biological filter. Once cycled, you will keep these in the tank indefinitely.
You will also have to get a bag to keep the biopellets in.
Fluval Edge Carbon Clean & Clear Renewal Sachet- 3 pack - Filter Media - Fish - PetSmart
But you can also find where some filters sell biopellets already in a bag (even better).
3. For physical filtration, buy some huge filter pads.
There are fine filter pads:
HBH Aqua-Pure Filter Media - Filter Media - Fish - PetSmart
And medium fine filter pads:
Marineland® Rite-Size Filter Pads - Filter Media - Fish - PetSmart
I would suggest getting both so that you can adjust how much physical filtration you want.
4. Forget carbon filter pads. Put some plants in the tank. Carbon removes some of the good things plants need.
This type of filter simply works by a fin spinning in the motor that is at the bottom of the filter. The intake is at the bottom front of the filter. The "pump" then simply pushes water up through a hole in the bottom of the "cup" that is the top half of the filter. The filter is designed with various pads that sit in the cup area that the water simply passes through (these are what I said to simply throw away). There is a thin mesh screen that sits at the front of the cup that is supposed to be your "cycled" media that you never change. Behind it is the physical filter pad filled with carbon. It is designed for you to spend $$$ changing this carbon filled filter pad freqently.
But instead, what you do is to cut out a piece of the medium fine filter pad to fit the shape of the filter "cup" and place that at the bottom of the cup where the water pumps out. Next, cut out a piece of the fine filter pad to fit the cup and place it on top of the medium fine filter pad. Next, place the biopellets in a bag on top of the filter pads. Start cycling your tank.
For maintained, simply cut out more pieces from the medium fine and fine filter pads and replace them as often as you want. You will keep your cycle with the bacteria that has colonized the biopellets.