Betta bulbs, well... some will sprout, some are duds. If you want to clean your tank every three days that's just fine. You can consider your tank 'uncycled' in that case, or in other words you have opted not to grow a colony of nitrifying bacteria. I suggest you read a bit on cycling anyways just because it is a useful thing to understand.
As far as plants go, you can have an uncycled planted tank, but you will really need to supplement with nutrients and ferts because the plants are gaining no nutritional value from the substrate. Furthermore, the plant's rooting systems may be disturbed from being constantly uprooted from the gravel when you clean your tank. Another thing to consider is the hassle involved in uprooting and replanting possibly dozens(?) of plants every time you clean your tank, which is frequently.
If you want to stick with your current setup, you would probably have some success with mosses, Java Fern, and Anubias. These plants do not depend heavily on root systems for nutrients (in fact, they do terribly when their roots are buried) and rely more on nutrition from the water, which you can easily provide with liquid ferts. Also, they are low-light plants so you lucky for you, you won't need to invest in some fancy fixture. It goes without saying that you will be dosing every water change, which would make me uneasy about all the cash you'll drop for ferts, but AHA, the tank is only a gallon a few drops should suffice. By tying anubias/moss/java fern on to driftwood, you can root it to a piece of driftwood for easy removal during water changes. Just plunk it in when the water is clean and VOILA-- instant aquascape every time!