Hi Lizzy... welcome, and thanks for answering my cat post, I checked that too!
Aren't gold bettas beautiful? I have 2 myself, a gold female & male breeding pair (I also breed some other colors, but I think golds are my favorite).
It's obvious that you know a little basic knowledge about the nitrogen cycle, so that's very good to see (also, the fact that you are concerned over small things that can affect your fish, and take the time out to try and correct them before they get out of hand). What I will tell you is the same as I told the others in the other betta posts... though bettas can be successfully kept in bowls with a number of IF's (if you keep the water from getting too cold, if you are religious about doing regular water changes & not overfeeding)... you will never get the full benefits of an-all-out nitrogen cycle as you would in a small tank with some kind of filtration. This is because the nitrogen cycle is pushed forward by bacteria that are primarily aerobic--such as species of Nitrosomas, which require some circulation to really grow at appreciable rates. In a bowl, you will eventually get small numbers of these living in the gravel as the small setup ages, but they will be low numbers, and so denitrify only at a very, very slow rate compared to that seen in a running filter. As a result, if you do decide to stick with a bowl, feeding becomes a very major issue. The way the manager at the pet store I work for sometimes puts it is "if you are a beginner and taking care of a betta, take how much food you think the betta needs, and feed it 1/4 of that". They will always look somewhat hungry because they are greedy, but overfeeding does a lot more harm than underfeeding in a mini-system that just can't handle the waste that is produced and not decomposed fast enough. This goes especially true for your case, where you are feeding bits of peas and bloodworms (both great foods in themselves, but they are frozen and so tend to dissipate into nutrient compounds in the water, decomposing quickly). So, for your case, I'd say:
1) Try feeding half as much
2) Your water changes sound great (really nice that you are willing to do small ones daily, that shows your devotion)
3) The nitrite spike sounds very temporary. In small bowls, it is extremely hard to acheive any kind of long-term balance, so you'll unfortunately have to deal with these from time to time if you decide to keep a bowl (I see nothing wrong with what you've done so far, and bettas are somewhat hardy fish that can deal with these ON OCCASION, if they are given a stable home otherwise, and if the flux goes back to normal within a short period of time, as yours sounds like it has already). However, for the long term, you might find it easier to manage a small 5 gallon setup which has a means of filtration and heat for the above reasons mentioned. If you should ever decide to change over, make sure you use all your old water, as this will help the tank cycle faster (your bacteria will pass over). Also, make any temperature adjustments slowly. Good luck whatever you decide...