You don't necessarily need a filter.

Biomedia in a filter will harbor BB (beneficial bacteria) that will convert deadly ammonia into relatively harmless nitrate. This is known as the nitrogen cycle. However, plants will also consume ammonia and BB will develop in your substrate and on all surfaces in your tank. If your bioload isn't high (one fish and one snail isn't much), your plants and BB can handle it. Monitor your ammonia and perform 20-30% weekly water changes by sucking out the detritus that collects on the bottom. This should be enough to keep your tank healthy.

Make sure that your plants get all the nutrients that they need. Your tap water and fish will provide most, but not all, of them. The Seachem Flourish will ensure complete coverage in your case. Dose it according to their directions. Filters won't remove these nutrients.

Filters also provide a side benefit of circulation, but you can easily supply this with pumps or aerators. Your tank is small enough that an aerator will do the job nicely. The idea is to try to move the water so that the plants get full access to all the nutrients and to disturb the water's surface so that good oxygen exchange can occur.

 

The only time a filter removes nutrients from the water column is when carbon is used. There are carbons that are plant friendly that leave nutrients alone.

As for whether a filter is needed or not, no, but since you are dealing with such a tiny amount of water, if something goes wrong, you are going to have problems if you don't stay on top of things like water changes.

Speaking of which, is your tank cycled? If not, you need to test your tank on daily basis until it is cycled. If it isn't, you need to do lots of water changes until it is. Ammonia is toxic and kills fish. The 2 plants you have are slow growers and the moss ball is actually a form of bacteria. It isn't going to be enough to convert ammonia in a non-cycled tank.

 

I'd do a small sponge filter for a tank that size, and as said above you'll need to be on top of water changes with slow-growing plants and a tiny tank. You can reduce your risk of ammonia spikes with the sponge filter harboring lots of BB, or you could even get a very small internal or hang on back filter and replace the carbon with ammonia absorbers.

 

Both your snail and betta can go to surface and breath air if needed. If you don’t overfeed and snail can clean up waste even better. Your gravel is a biological filter once it gets cycled. Moss balls are actually very good water filters all on their own but actually do a little better job if they have at least a little circulation. If you do go filterless make sure water surface stays pretty film free, if necessary take a paper towel piece and lay on surface for couple seconds and then pull of scum layer or do your best to get as much as you can at water change.

I’d even get a small colony red root floaters or salvia and maintain them covering 1/4 of water surface, not enough to severely block out light to plants below. Floating plants have access to all co2/light they want at water surface and can process a lot pollutants from water as well as exchange a lot of gases. Maybe even 2-3 pieces of frogbit, you’d probably like look of those longer roots dangling in water and your betta will love it.

Some of bigger Anubis etc can be maintained with tips of leaves poking out of water surface. Emergent plants again can grab co2 and other gases from air and pass them down to roots in gravel bed and also get higher light at surface. Also a good water filter.

This was a great pic I saw today of using emergent plants in a no filter micro vase.

Wonderful thread of his tanks.
https://www.plantedtank.net/forums/...forums/12-tank-journals/731794-xzs-3ft-high-tech-low-tech-nano-experiments.html

 

Snails are fine, but they also poop which contributes to the bio load of the the tank. This needs to be kept in mind when it comes to tank maintenance. Tiny tanks when they have fauna or fish are a lot more work than larger tanks because of the volume of water that they have as I've already mentioned.

Something I didn't mention yesterday was ferts. For a tank that size, only a drop or two should be used and it shouldn't be weekly. If there is too much added, the tank is going to turn algae farm faster than a larger tank. Since the tank only contains slow growers, dumping in a bunch of ferts isn't going to change how fast they grow. This is also where water testing comes in handy until it is figured out how much and how often plants should be fed. If a tank already has 20% nitrates, it doesn't need more. For a tank that size and with that many plants, I wouldn't allow nitrates to get that high. I'd keep it at around 5%.

 

Here's a 5 gallon Walstadt type that the owner has Sparkling Gouramies and NeoCaridinas breeding in. No filter, just a heater and a light fixture.